Theorem ordunifi 7788

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 6618	. . . . . 6 |- _E We On
2		weso 4879	. . . . . 6 |- ( _E We On -> _E Or On )
3	1, 2	ax-mp 5	. . . . 5 |- _E Or On
4		soss 4827	. . . . 5 |- ( A C_ On -> ( _E Or On -> _E Or A ) )
5	3, 4	mpi 17	. . . 4 |- ( A C_ On -> _E Or A )
6		fimax2g 7784	. . . 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 3494	. . . . . . . . 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 3494	. . . . . . . . 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 4921	. . . . . . . . 9 |- ( ( y e. On /\ x e. On ) -> ( y C_ x <-> -. x e. y ) )
13		epel 4803	. . . . . . . . . 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 2893	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> A. y e. A y C_ x ) )
18		unissb 4283	. . . . . 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 2965	. . . 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 4281	. . . 4 |- ( x e. A -> x C_ U. A )
24		eqss 3514	. . . . 5 |- ( x = U. A <-> ( x C_ U. A /\ U. A C_ x ) )
25		eleq1 2529	. . . . . 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 2943	. 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 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   E.wrex 2808   C_ wss 3471   (/)c0 3793   U.cuni 4251   class class class wbr 4456   _E cep 4798   Or wor 4808   We wwe 4846   Oncon0 4887   Fincfn 7535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-er 7329  df-en 7536  df-fin 7539

This theorem is referenced by:  nnunifi  7789  oemapvali  8120  ttukeylem6  8911  limsucncmpi  30115