Theorem iinon 3910

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 3012	. . . 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 1650	. . . . . . 7 |- (E.x e. A y = B <-> E.x(x e. A /\ y = B))
3	2	exbii 1051	. . . . . 6 |- (E.yE.x e. A y = B <-> E.yE.x(x e. A /\ y = B))
4		excom 1046	. . . . . 6 |- (E.xE.y(x e. A /\ y = B) <-> E.yE.x(x e. A /\ y = B))
5		19.42v 1308	. . . . . . . 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 1815	. . . . . . . 8 |- E.y y = B
8	5, 7	mpbiran2 729	. . . . . . 7 |- (E.y(x e. A /\ y = B) <-> x e. A)
9	8	exbii 1051	. . . . . 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 2288	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
12		abn0 2290	. . . . 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 1687	. . . . . . 7 |- (A.x e. A B e. On -> A.xA.x e. A B e. On)
16		ax-17 971	. . . . . . 7 |- (y e. On -> A.x y e. On)
17		ra4 1694	. . . . . . . 8 |- (A.x e. A B e. On -> (x e. A -> B e. On))
18		eleq1a 1543	. . . . . . . 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 1745	. . . . . 6 |- (A.x e. A B e. On -> (E.x e. A y = B -> y e. On))
21		abid 1465	. . . . . 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 1286	. . . 4 |- (A.x e. A B e. On -> A.y(y e. {y | E.x e. A y = B} -> y e. On))
24		hbab1 1466	. . . . 5 |- (z e. {y | E.x e. A y = B} -> A.y z e. {y | E.x e. A y = B})
25		ax-17 971	. . . . 5 |- (z e. On -> A.y z e. On)
26	24, 25	dfss2f 2060	. . . 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 2588	. 2 |- |^|_x e. A B = |^|{y | E.x e. A y = B}
30	28, 29	syl5eqel 1552	1 |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  |^|cint 2533  |^|_ciin 2567  Oncon0 2948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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