Theorem iinon 3901

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 3007	. . . 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 1647	. . . . . . 7 |- (E.x e. A y = B <-> E.x(x e. A /\ y = B))
3	2	exbii 1049	. . . . . 6 |- (E.yE.x e. A y = B <-> E.yE.x(x e. A /\ y = B))
4		excom 1044	. . . . . 6 |- (E.xE.y(x e. A /\ y = B) <-> E.yE.x(x e. A /\ y = B))
5		19.42v 1306	. . . . . . . 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 1811	. . . . . . . 8 |- E.y y = B
8	5, 7	mpbiran2 728	. . . . . . 7 |- (E.y(x e. A /\ y = B) <-> x e. A)
9	8	exbii 1049	. . . . . 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 2284	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
12		abn0 2286	. . . . 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 452	. . 3 |- (({y | E.x e. A y = B} (_ On /\ A =/= (/)) -> |^|{y | E.x e. A y = B} e. On)
15		hbra1 1684	. . . . . . 7 |- (A.x e. A B e. On -> A.xA.x e. A B e. On)
16		ax-17 969	. . . . . . 7 |- (y e. On -> A.x y e. On)
17		ra4 1691	. . . . . . . 8 |- (A.x e. A B e. On -> (x e. A -> B e. On))
18		eleq1a 1540	. . . . . . . 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 1742	. . . . . 6 |- (A.x e. A B e. On -> (E.x e. A y = B -> y e. On))
21		abid 1463	. . . . . 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 1284	. . . 4 |- (A.x e. A B e. On -> A.y(y e. {y | E.x e. A y = B} -> y e. On))
24		hbab1 1464	. . . . 5 |- (z e. {y | E.x e. A y = B} -> A.y z e. {y | E.x e. A y = B})
25		ax-17 969	. . . . 5 |- (z e. On -> A.y z e. On)
26	24, 25	dfss2f 2056	. . . 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 448	. 2 |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|{y | E.x e. A y = B} e. On)
29	6	dfiin2 2583	. 2 |- |^|_x e. A B = |^|{y | E.x e. A y = B}
30	28, 29	syl5eqel 1549	1 |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807   (_ wss 2043  (/)c0 2276  |^|cint 2528  |^|_ciin 2562  Oncon0 2943

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iin 2564  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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