Theorem onminex 3026

Description: If a wff is true for an ordinal number, there is a smallest ordinal number for which it is true.

Hypothesis

Ref	Expression
onminex.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
onminex	|- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))

Distinct variable groups:   x,y   ph,y   ps,x

Proof of Theorem onminex

Step	Hyp	Ref	Expression
1		hbab1 1469	. . . . . . 7 |- (y e. {x | (x e. On /\ ph)} -> A.x y e. {x | (x e. On /\ ph)})
2	1	hbint 2547	. . . . . 6 |- (y e. |^|{x | (x e. On /\ ph)} -> A.x y e. |^|{x | (x e. On /\ ph)})
3	2, 1	hbel 1569	. . . . . . 7 |- (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} -> A.x|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
4		ax-17 973	. . . . . . . . 9 |- (-. ps -> A.x -. ps)
5	2, 4	hbim 1009	. . . . . . . 8 |- ((y e. |^|{x | (x e. On /\ ph)} -> -. ps) -> A.x(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
6	5	hbal 1007	. . . . . . 7 |- (A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps) -> A.xA.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
7	3, 6	hban 1011	. . . . . 6 |- ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> A.x(|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
8		eleq1 1537	. . . . . . 7 |- (x = |^|{x | (x e. On /\ ph)} -> (x e. {x | (x e. On /\ ph)} <-> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)}))
9		eleq2 1538	. . . . . . . . 9 |- (x = |^|{x | (x e. On /\ ph)} -> (y e. x <-> y e. |^|{x | (x e. On /\ ph)}))
10	9	imbi1d 615	. . . . . . . 8 |- (x = |^|{x | (x e. On /\ ph)} -> ((y e. x -> -. ps) <-> (y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
11	10	albidv 1280	. . . . . . 7 |- (x = |^|{x | (x e. On /\ ph)} -> (A.y(y e. x -> -. ps) <-> A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
12	8, 11	anbi12d 630	. . . . . 6 |- (x = |^|{x | (x e. On /\ ph)} -> ((x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))))
13	2, 7, 12	cla4egf 1864	. . . . 5 |- (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} -> ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps))))
14	13	anabsi5 497	. . . 4 |- ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
15		ssab2 2133	. . . . 5 |- {x | (x e. On /\ ph)} (_ On
16		onint 3012	. . . . 5 |- (({x | (x e. On /\ ph)} (_ On /\ {x | (x e. On /\ ph)} =/= (/)) -> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
17	15, 16	mpan 697	. . . 4 |- ({x | (x e. On /\ ph)} =/= (/) -> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
18		oninton 3018	. . . . . . . 8 |- (({x | (x e. On /\ ph)} (_ On /\ {x | (x e. On /\ ph)} =/= (/)) -> |^|{x | (x e. On /\ ph)} e. On)
19	15, 18	mpan 697	. . . . . . 7 |- ({x | (x e. On /\ ph)} =/= (/) -> |^|{x | (x e. On /\ ph)} e. On)
20		onelon 2978	. . . . . . . 8 |- ((|^|{x | (x e. On /\ ph)} e. On /\ y e. |^|{x | (x e. On /\ ph)}) -> y e. On)
21	20	ex 373	. . . . . . 7 |- (|^|{x | (x e. On /\ ph)} e. On -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
22	19, 21	syl 10	. . . . . 6 |- ({x | (x e. On /\ ph)} =/= (/) -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
23		visset 1816	. . . . . . . . . 10 |- y e. V
24		eleq1 1537	. . . . . . . . . . 11 |- (x = y -> (x e. On <-> y e. On))
25		onminex.1	. . . . . . . . . . 11 |- (x = y -> (ph <-> ps))
26	24, 25	anbi12d 630	. . . . . . . . . 10 |- (x = y -> ((x e. On /\ ph) <-> (y e. On /\ ps)))
27	23, 26	elab 1900	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} <-> (y e. On /\ ps))
28		onnmin 3021	. . . . . . . . . 10 |- (({x | (x e. On /\ ph)} (_ On /\ y e. {x | (x e. On /\ ph)}) -> -. y e. |^|{x | (x e. On /\ ph)})
29	15, 28	mpan 697	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} -> -. y e. |^|{x | (x e. On /\ ph)})
30	27, 29	sylbir 201	. . . . . . . 8 |- ((y e. On /\ ps) -> -. y e. |^|{x | (x e. On /\ ph)})
31	30	ex 373	. . . . . . 7 |- (y e. On -> (ps -> -. y e. |^|{x | (x e. On /\ ph)}))
32	31	con2d 91	. . . . . 6 |- (y e. On -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
33	22, 32	syli 54	. . . . 5 |- ({x | (x e. On /\ ph)} =/= (/) -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
34	33	19.21aiv 1288	. . . 4 |- ({x | (x e. On /\ ph)} =/= (/) -> A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
35	14, 17, 34	sylanc 473	. . 3 |- ({x | (x e. On /\ ph)} =/= (/) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
36		abn0 2294	. . 3 |- ({x | (x e. On /\ ph)} =/= (/) <-> E.x(x e. On /\ ph))
37		abid 1468	. . . . . . 7 |- (x e. {x | (x e. On /\ ph)} <-> (x e. On /\ ph))
38	37	bicomi 172	. . . . . 6 |- ((x e. On /\ ph) <-> x e. {x | (x e. On /\ ph)})
39		df-ral 1652	. . . . . 6 |- (A.y e. x -. ps <-> A.y(y e. x -> -. ps))
40	38, 39	anbi12i 484	. . . . 5 |- (((x e. On /\ ph) /\ A.y e. x -. ps) <-> (x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
41		anass 441	. . . . 5 |- (((x e. On /\ ph) /\ A.y e. x -. ps) <-> (x e. On /\ (ph /\ A.y e. x -. ps)))
42	40, 41	bitr3 175	. . . 4 |- ((x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> (x e. On /\ (ph /\ A.y e. x -. ps)))
43	42	exbii 1053	. . 3 |- (E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
44	35, 36, 43	3imtr3 218	. 2 |- (E.x(x e. On /\ ph) -> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
45		df-rex 1653	. 2 |- (E.x e. On ph <-> E.x(x e. On /\ ph))
46		df-rex 1653	. 2 |- (E.x e. On (ph /\ A.y e. x -. ps) <-> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
47	44, 45, 46	3imtr4 219	1 |- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   =/= wne 1588  A.wral 1648  E.wrex 1649   (_ wss 2050  (/)c0 2283  |^|cint 2537  Oncon0 2954

This theorem is referenced by:  tz7.49 3965  zorn2lem7 4804

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846