Theorem onint 6995

Description: The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)

Assertion

Ref	Expression
onint	|- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. A )

Proof of Theorem onint

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordon 6982	. . . 4 |- Ord On
2		tz7.5 5744	. . . 4 |- ( ( Ord On /\ A C_ On /\ A =/= (/) ) -> E. x e. A ( A i^i x ) = (/) )
3	1, 2	mp3an1 1411	. . 3 |- ( ( A C_ On /\ A =/= (/) ) -> E. x e. A ( A i^i x ) = (/) )
4		ssel 3597	. . . . . . . . . . . . . . . 16 |- ( A C_ On -> ( x e. A -> x e. On ) )
5	4	imdistani 726	. . . . . . . . . . . . . . 15 |- ( ( A C_ On /\ x e. A ) -> ( A C_ On /\ x e. On ) )
6		ssel 3597	. . . . . . . . . . . . . . . . . . . 20 |- ( A C_ On -> ( z e. A -> z e. On ) )
7		ontri1 5757	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. On /\ z e. On ) -> ( x C_ z <-> -. z e. x ) )
8		ssel 3597	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x C_ z -> ( y e. x -> y e. z ) )
9	7, 8	syl6bir 244	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. On /\ z e. On ) -> ( -. z e. x -> ( y e. x -> y e. z ) ) )
10	9	ex 450	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. On -> ( z e. On -> ( -. z e. x -> ( y e. x -> y e. z ) ) ) )
11	6, 10	sylan9 689	. . . . . . . . . . . . . . . . . . 19 |- ( ( A C_ On /\ x e. On ) -> ( z e. A -> ( -. z e. x -> ( y e. x -> y e. z ) ) ) )
12	11	com4r 94	. . . . . . . . . . . . . . . . . 18 |- ( y e. x -> ( ( A C_ On /\ x e. On ) -> ( z e. A -> ( -. z e. x -> y e. z ) ) ) )
13	12	imp31 448	. . . . . . . . . . . . . . . . 17 |- ( ( ( y e. x /\ ( A C_ On /\ x e. On ) ) /\ z e. A ) -> ( -. z e. x -> y e. z ) )
14	13	ralimdva 2962	. . . . . . . . . . . . . . . 16 |- ( ( y e. x /\ ( A C_ On /\ x e. On ) ) -> ( A. z e. A -. z e. x -> A. z e. A y e. z ) )
15		disj 4017	. . . . . . . . . . . . . . . 16 |- ( ( A i^i x ) = (/) <-> A. z e. A -. z e. x )
16		vex 3203	. . . . . . . . . . . . . . . . 17 |- y e. _V
17	16	elint2 4482	. . . . . . . . . . . . . . . 16 |- ( y e. |^| A <-> A. z e. A y e. z )
18	14, 15, 17	3imtr4g 285	. . . . . . . . . . . . . . 15 |- ( ( y e. x /\ ( A C_ On /\ x e. On ) ) -> ( ( A i^i x ) = (/) -> y e. |^| A ) )
19	5, 18	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( y e. x /\ ( A C_ On /\ x e. A ) ) -> ( ( A i^i x ) = (/) -> y e. |^| A ) )
20	19	exp32 631	. . . . . . . . . . . . 13 |- ( y e. x -> ( A C_ On -> ( x e. A -> ( ( A i^i x ) = (/) -> y e. |^| A ) ) ) )
21	20	com4l 92	. . . . . . . . . . . 12 |- ( A C_ On -> ( x e. A -> ( ( A i^i x ) = (/) -> ( y e. x -> y e. |^| A ) ) ) )
22	21	imp32 449	. . . . . . . . . . 11 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> ( y e. x -> y e. |^| A ) )
23	22	ssrdv 3609	. . . . . . . . . 10 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> x C_ |^| A )
24		intss1 4492	. . . . . . . . . . 11 |- ( x e. A -> |^| A C_ x )
25	24	ad2antrl 764	. . . . . . . . . 10 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> |^| A C_ x )
26	23, 25	eqssd 3620	. . . . . . . . 9 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> x = |^| A )
27	26	eleq1d 2686	. . . . . . . 8 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> ( x e. A <-> |^| A e. A ) )
28	27	biimpd 219	. . . . . . 7 |- ( ( A C_ On /\ ( x e. A /\ ( A i^i x ) = (/) ) ) -> ( x e. A -> |^| A e. A ) )
29	28	exp32 631	. . . . . 6 |- ( A C_ On -> ( x e. A -> ( ( A i^i x ) = (/) -> ( x e. A -> |^| A e. A ) ) ) )
30	29	com34 91	. . . . 5 |- ( A C_ On -> ( x e. A -> ( x e. A -> ( ( A i^i x ) = (/) -> |^| A e. A ) ) ) )
31	30	pm2.43d 53	. . . 4 |- ( A C_ On -> ( x e. A -> ( ( A i^i x ) = (/) -> |^| A e. A ) ) )
32	31	rexlimdv 3030	. . 3 |- ( A C_ On -> ( E. x e. A ( A i^i x ) = (/) -> |^| A e. A ) )
33	3, 32	syl5 34	. 2 |- ( A C_ On -> ( ( A C_ On /\ A =/= (/) ) -> |^| A e. A ) )
34	33	anabsi5 858	1 |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|cint 4475   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  onint0  6996  onssmin  6997  onminesb  6998  onminsb  6999  oninton  7000  oneqmin  7005  oeeulem  7681  nnawordex  7717  unblem1  8212  unblem2  8213  tz9.12lem3  8652  scott0  8749  cardid2  8779  ackbij1lem18  9059  cardcf  9074  cff1  9080  cflim2  9085  cfss  9087  cofsmo  9091  fin23lem26  9147  pwfseqlem3  9482  gruina  9640  2ndcdisj  21259  sltval2  31809  nocvxmin  31894  rankeq1o  32278  dnnumch3  37617