Theorem ordunifi 8210

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 6983	. . . . . 6 |- _E We On
2		weso 5105	. . . . . 6 |- ( _E We On -> _E Or On )
3	1, 2	ax-mp 5	. . . . 5 |- _E Or On
4		soss 5053	. . . . 5 |- ( A C_ On -> ( _E Or On -> _E Or A ) )
5	3, 4	mpi 20	. . . 4 |- ( A C_ On -> _E Or A )
6		fimax2g 8206	. . . 4 |- ( ( _E Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
7	5, 6	syl3an1 1359	. . 3 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
8		ssel2 3598	. . . . . . . . 9 |- ( ( A C_ On /\ y e. A ) -> y e. On )
9	8	adantlr 751	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> y e. On )
10		ssel2 3598	. . . . . . . . 9 |- ( ( A C_ On /\ x e. A ) -> x e. On )
11	10	adantr 481	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> x e. On )
12		ontri1 5757	. . . . . . . . 9 |- ( ( y e. On /\ x e. On ) -> ( y C_ x <-> -. x e. y ) )
13		epel 5032	. . . . . . . . . 10 |- ( x _E y <-> x e. y )
14	13	notbii 310	. . . . . . . . 9 |- ( -. x _E y <-> -. x e. y )
15	12, 14	syl6rbbr 279	. . . . . . . 8 |- ( ( y e. On /\ x e. On ) -> ( -. x _E y <-> y C_ x ) )
16	9, 11, 15	syl2anc 693	. . . . . . 7 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> ( -. x _E y <-> y C_ x ) )
17	16	ralbidva 2985	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> A. y e. A y C_ x ) )
18		unissb 4469	. . . . . 6 |- ( U. A C_ x <-> A. y e. A y C_ x )
19	17, 18	syl6bbr 278	. . . . 5 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> U. A C_ x ) )
20	19	rexbidva 3049	. . . 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 1082	. . 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 222	. 2 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A U. A C_ x )
23		elssuni 4467	. . . 4 |- ( x e. A -> x C_ U. A )
24		eqss 3618	. . . . 5 |- ( x = U. A <-> ( x C_ U. A /\ U. A C_ x ) )
25		eleq1 2689	. . . . . 6 |- ( x = U. A -> ( x e. A <-> U. A e. A ) )
26	25	biimpcd 239	. . . . 5 |- ( x e. A -> ( x = U. A -> U. A e. A ) )
27	24, 26	syl5bir 233	. . . 4 |- ( x e. A -> ( ( x C_ U. A /\ U. A C_ x ) -> U. A e. A ) )
28	23, 27	mpand 711	. . 3 |- ( x e. A -> ( U. A C_ x -> U. A e. A ) )
29	28	rexlimiv 3027	. 2 |- ( E. x e. A U. A C_ x -> U. A e. A )
30	22, 29	syl 17	1 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   _E cep 5028   Or wor 5034   We wwe 5072   Oncon0 5723   Fincfn 7955

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-pow 4843  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-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959

This theorem is referenced by:  nnunifi  8211  oemapvali  8581  ttukeylem6  9336  limsucncmpi  32444