Theorem iinon 3916

Description: The nonempty indexed intersection of a class of ordinal numbers B(x) is an ordinal number.

Hypothesis

Ref	Expression
iinon.1	|- B e. V

Assertion

Ref	Expression
iinon	|- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)

Distinct variable group:   x,A

Proof of Theorem iinon

Step	Hyp	Ref	Expression
1		oninton 3018	. . . 4 |- (({y | E.x e. A y = B} (_ On /\ {y | E.x e. A y = B} =/= (/)) -> |^|{y | E.x e. A y = B} e. On)
2		df-rex 1653	. . . . . . 7 |- (E.x e. A y = B <-> E.x(x e. A /\ y = B))
3	2	exbii 1053	. . . . . 6 |- (E.yE.x e. A y = B <-> E.yE.x(x e. A /\ y = B))
4		excom 1048	. . . . . 6 |- (E.xE.y(x e. A /\ y = B) <-> E.yE.x(x e. A /\ y = B))
5		19.42v 1310	. . . . . . . 8 |- (E.y(x e. A /\ y = B) <-> (x e. A /\ E.y y = B))
6		iinon.1	. . . . . . . . 9 |- B e. V
7	6	isseti 1818	. . . . . . . 8 |- E.y y = B
8	5, 7	mpbiran2 731	. . . . . . 7 |- (E.y(x e. A /\ y = B) <-> x e. A)
9	8	exbii 1053	. . . . . 6 |- (E.xE.y(x e. A /\ y = B) <-> E.x x e. A)
10	3, 4, 9	3bitr2r 180	. . . . 5 |- (E.x x e. A <-> E.yE.x e. A y = B)
11		ne0 2292	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
12		abn0 2294	. . . . 5 |- ({y | E.x e. A y = B} =/= (/) <-> E.yE.x e. A y = B)
13	10, 11, 12	3bitr4 183	. . . 4 |- (A =/= (/) <-> {y | E.x e. A y = B} =/= (/))
14	1, 13	sylan2b 454	. . 3 |- (({y | E.x e. A y = B} (_ On /\ A =/= (/)) -> |^|{y | E.x e. A y = B} e. On)
15		hbra1 1690	. . . . . . 7 |- (A.x e. A B e. On -> A.xA.x e. A B e. On)
16		ax-17 973	. . . . . . 7 |- (y e. On -> A.x y e. On)
17		ra4 1697	. . . . . . . 8 |- (A.x e. A B e. On -> (x e. A -> B e. On))
18		eleq1a 1546	. . . . . . . 8 |- (B e. On -> (y = B -> y e. On))
19	17, 18	syl6 22	. . . . . . 7 |- (A.x e. A B e. On -> (x e. A -> (y = B -> y e. On)))
20	15, 16, 19	r19.23ad 1748	. . . . . 6 |- (A.x e. A B e. On -> (E.x e. A y = B -> y e. On))
21		abid 1468	. . . . . 6 |- (y e. {y | E.x e. A y = B} <-> E.x e. A y = B)
22	20, 21	syl5ib 206	. . . . 5 |- (A.x e. A B e. On -> (y e. {y | E.x e. A y = B} -> y e. On))
23	22	19.21aiv 1288	. . . 4 |- (A.x e. A B e. On -> A.y(y e. {y | E.x e. A y = B} -> y e. On))
24		hbab1 1469	. . . . 5 |- (z e. {y | E.x e. A y = B} -> A.y z e. {y | E.x e. A y = B})
25		ax-17 973	. . . . 5 |- (z e. On -> A.y z e. On)
26	24, 25	dfss2f 2063	. . . 4 |- ({y | E.x e. A y = B} (_ On <-> A.y(y e. {y | E.x e. A y = B} -> y e. On))
27	23, 26	sylibr 200	. . 3 |- (A.x e. A B e. On -> {y | E.x e. A y = B} (_ On)
28	14, 27	sylan 450	. 2 |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|{y | E.x e. A y = B} e. On)
29	6	dfiin2 2592	. 2 |- |^|_x e. A B = |^|{y | E.x e. A y = B}
30	28, 29	syl5eqel 1555	1 |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   =/= wne 1588  A.wral 1648  E.wrex 1649  Vcvv 1814   (_ wss 2050  (/)c0 2283  |^|cint 2537  |^|_ciin 2571  Oncon0 2954

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-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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