Theorem onmindif2 3051

Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed.

Assertion

Ref	Expression
onmindif2	|- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Proof of Theorem onmindif2

Step	Hyp	Ref	Expression
1		onnmin 3005	. . . . . . . . . . 11 |- ((A (_ On /\ x e. A) -> -. x e. |^|A)
2	1	adantlr 393	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> -. x e. |^|A)
3		ontri1 2971	. . . . . . . . . . . 12 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> -. x e. |^|A))
4		onsseleq 2989	. . . . . . . . . . . 12 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> (|^|A e. x \/ |^|A = x)))
5	3, 4	bitr3d 528	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
6		oninton 3002	. . . . . . . . . . . 12 |- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
7	6	adantr 389	. . . . . . . . . . 11 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> |^|A e. On)
8		ssel2 2054	. . . . . . . . . . . 12 |- ((A (_ On /\ x e. A) -> x e. On)
9	8	adantlr 393	. . . . . . . . . . 11 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> x e. On)
10	5, 7, 9	sylanc 471	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
11	2, 10	mpbid 195	. . . . . . . . 9 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (|^|A e. x \/ |^|A = x))
12	11	ord 232	. . . . . . . 8 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> |^|A = x))
13		eqcom 1469	. . . . . . . 8 |- (|^|A = x <-> x = |^|A)
14	12, 13	syl6ib 212	. . . . . . 7 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> x = |^|A))
15	14	necon1ad 1623	. . . . . 6 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (x =/= |^|A -> |^|A e. x))
16	15	ex 373	. . . . 5 |- ((A (_ On /\ A =/= (/)) -> (x e. A -> (x =/= |^|A -> |^|A e. x)))
17	16	imp3a 361	. . . 4 |- ((A (_ On /\ A =/= (/)) -> ((x e. A /\ x =/= |^|A) -> |^|A e. x))
18		eldifsn 2453	. . . 4 |- (x e. (A \ {|^|A}) <-> (x e. A /\ x =/= |^|A))
19	17, 18	syl5ib 206	. . 3 |- ((A (_ On /\ A =/= (/)) -> (x e. (A \ {|^|A}) -> |^|A e. x))
20	19	r19.21aiv 1705	. 2 |- ((A (_ On /\ A =/= (/)) -> A.x e. (A \ {|^|A})|^|A e. x)
21		intex 2719	. . . 4 |- (A =/= (/) <-> |^|A e. V)
22		elintg 2531	. . . 4 |- (|^|A e. V -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
23	21, 22	sylbi 199	. . 3 |- (A =/= (/) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
24	23	adantl 388	. 2 |- ((A (_ On /\ A =/= (/)) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
25	20, 24	mpbird 196	1 |- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  Vcvv 1802   \ cdif 2034   (_ wss 2037  (/)c0 2270  {csn 2399  |^|cint 2523  Oncon0 2938

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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