Theorem ordunifi 7567

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 6400	. . . . . 6 |- _E We On
2		weso 4716	. . . . . 6 |- ( _E We On -> _E Or On )
3	1, 2	ax-mp 5	. . . . 5 |- _E Or On
4		soss 4664	. . . . 5 |- ( A C_ On -> ( _E Or On -> _E Or A ) )
5	3, 4	mpi 17	. . . 4 |- ( A C_ On -> _E Or A )
6		fimax2g 7563	. . . 4 |- ( ( _E Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
7	5, 6	syl3an1 1251	. . 3 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
8		ssel2 3356	. . . . . . . . 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 3356	. . . . . . . . 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 4758	. . . . . . . . 9 |- ( ( y e. On /\ x e. On ) -> ( y C_ x <-> -. x e. y ) )
13		epel 4640	. . . . . . . . . 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 2736	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> A. y e. A y C_ x ) )
18		unissb 4128	. . . . . 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 2737	. . . 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 1009	. . 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 4126	. . . 4 |- ( x e. A -> x C_ U. A )
24		eqss 3376	. . . . 5 |- ( x = U. A <-> ( x C_ U. A /\ U. A C_ x ) )
25		eleq1 2503	. . . . . 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 2840	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2611   A.wral 2720   E.wrex 2721   C_ wss 3333   (/)c0 3642   U.cuni 4096   class class class wbr 4297   _E cep 4635   Or wor 4645   We wwe 4683   Oncon0 4724   Fincfn 7315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-1o 6925  df-er 7106  df-en 7316  df-fin 7319

This theorem is referenced by:  nnunifi  7568  oemapvali  7897  ttukeylem6  8688  limsucncmpi  28296