Theorem ordunifi 7770

Description: The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Assertion

Ref	Expression
ordunifi	|- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> U. A e. A )

Proof of Theorem ordunifi

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

Step	Hyp	Ref	Expression
1		epweon 6603	. . . . . 6 |- _E We On
2		weso 4870	. . . . . 6 |- ( _E We On -> _E Or On )
3	1, 2	ax-mp 5	. . . . 5 |- _E Or On
4		soss 4818	. . . . 5 |- ( A C_ On -> ( _E Or On -> _E Or A ) )
5	3, 4	mpi 17	. . . 4 |- ( A C_ On -> _E Or A )
6		fimax2g 7766	. . . 4 |- ( ( _E Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
7	5, 6	syl3an1 1261	. . 3 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
8		ssel2 3499	. . . . . . . . 9 |- ( ( A C_ On /\ y e. A ) -> y e. On )
9	8	adantlr 714	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> y e. On )
10		ssel2 3499	. . . . . . . . 9 |- ( ( A C_ On /\ x e. A ) -> x e. On )
11	10	adantr 465	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> x e. On )
12		ontri1 4912	. . . . . . . . 9 |- ( ( y e. On /\ x e. On ) -> ( y C_ x <-> -. x e. y ) )
13		epel 4794	. . . . . . . . . 10 |- ( x _E y <-> x e. y )
14	13	notbii 296	. . . . . . . . 9 |- ( -. x _E y <-> -. x e. y )
15	12, 14	syl6rbbr 264	. . . . . . . 8 |- ( ( y e. On /\ x e. On ) -> ( -. x _E y <-> y C_ x ) )
16	9, 11, 15	syl2anc 661	. . . . . . 7 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> ( -. x _E y <-> y C_ x ) )
17	16	ralbidva 2900	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> A. y e. A y C_ x ) )
18		unissb 4277	. . . . . 6 |- ( U. A C_ x <-> A. y e. A y C_ x )
19	17, 18	syl6bbr 263	. . . . 5 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> U. A C_ x ) )
20	19	rexbidva 2970	. . . 4 |- ( A C_ On -> ( E. x e. A A. y e. A -. x _E y <-> E. x e. A U. A C_ x ) )
21	20	3ad2ant1 1017	. . 3 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> ( E. x e. A A. y e. A -. x _E y <-> E. x e. A U. A C_ x ) )
22	7, 21	mpbid 210	. 2 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A U. A C_ x )
23		elssuni 4275	. . . 4 |- ( x e. A -> x C_ U. A )
24		eqss 3519	. . . . 5 |- ( x = U. A <-> ( x C_ U. A /\ U. A C_ x ) )
25		eleq1 2539	. . . . . 6 |- ( x = U. A -> ( x e. A <-> U. A e. A ) )
26	25	biimpcd 224	. . . . 5 |- ( x e. A -> ( x = U. A -> U. A e. A ) )
27	24, 26	syl5bir 218	. . . 4 |- ( x e. A -> ( ( x C_ U. A /\ U. A C_ x ) -> U. A e. A ) )
28	23, 27	mpand 675	. . 3 |- ( x e. A -> ( U. A C_ x -> U. A e. A ) )
29	28	rexlimiv 2949	. 2 |- ( E. x e. A U. A C_ x -> U. A e. A )
30	22, 29	syl 16	1 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   C_ wss 3476   (/)c0 3785   U.cuni 4245   class class class wbr 4447   _E cep 4789   Or wor 4799   We wwe 4837   Oncon0 4878   Fincfn 7516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-er 7311  df-en 7517  df-fin 7520

This theorem is referenced by:  nnunifi  7771  oemapvali  8103  ttukeylem6  8894  limsucncmpi  29515