Theorem ordunifi 7821

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 6610	. . . . . 6 |- _E We On
2		weso 4825	. . . . . 6 |- ( _E We On -> _E Or On )
3	1, 2	ax-mp 5	. . . . 5 |- _E Or On
4		soss 4773	. . . . 5 |- ( A C_ On -> ( _E Or On -> _E Or A ) )
5	3, 4	mpi 20	. . . 4 |- ( A C_ On -> _E Or A )
6		fimax2g 7817	. . . 4 |- ( ( _E Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
7	5, 6	syl3an1 1301	. . 3 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A -. x _E y )
8		ssel2 3427	. . . . . . . . 9 |- ( ( A C_ On /\ y e. A ) -> y e. On )
9	8	adantlr 721	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> y e. On )
10		ssel2 3427	. . . . . . . . 9 |- ( ( A C_ On /\ x e. A ) -> x e. On )
11	10	adantr 467	. . . . . . . 8 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> x e. On )
12		ontri1 5457	. . . . . . . . 9 |- ( ( y e. On /\ x e. On ) -> ( y C_ x <-> -. x e. y ) )
13		epel 4748	. . . . . . . . . 10 |- ( x _E y <-> x e. y )
14	13	notbii 298	. . . . . . . . 9 |- ( -. x _E y <-> -. x e. y )
15	12, 14	syl6rbbr 268	. . . . . . . 8 |- ( ( y e. On /\ x e. On ) -> ( -. x _E y <-> y C_ x ) )
16	9, 11, 15	syl2anc 667	. . . . . . 7 |- ( ( ( A C_ On /\ x e. A ) /\ y e. A ) -> ( -. x _E y <-> y C_ x ) )
17	16	ralbidva 2824	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> A. y e. A y C_ x ) )
18		unissb 4229	. . . . . 6 |- ( U. A C_ x <-> A. y e. A y C_ x )
19	17, 18	syl6bbr 267	. . . . 5 |- ( ( A C_ On /\ x e. A ) -> ( A. y e. A -. x _E y <-> U. A C_ x ) )
20	19	rexbidva 2898	. . . 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 1029	. . 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 214	. 2 |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> E. x e. A U. A C_ x )
23		elssuni 4227	. . . 4 |- ( x e. A -> x C_ U. A )
24		eqss 3447	. . . . 5 |- ( x = U. A <-> ( x C_ U. A /\ U. A C_ x ) )
25		eleq1 2517	. . . . . 6 |- ( x = U. A -> ( x e. A <-> U. A e. A ) )
26	25	biimpcd 228	. . . . 5 |- ( x e. A -> ( x = U. A -> U. A e. A ) )
27	24, 26	syl5bir 222	. . . 4 |- ( x e. A -> ( ( x C_ U. A /\ U. A C_ x ) -> U. A e. A ) )
28	23, 27	mpand 681	. . 3 |- ( x e. A -> ( U. A C_ x -> U. A e. A ) )
29	28	rexlimiv 2873	. 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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   C_ wss 3404   (/)c0 3731   U.cuni 4198   class class class wbr 4402   _E cep 4743   Or wor 4754   We wwe 4792   Oncon0 5423   Fincfn 7569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-er 7363  df-en 7570  df-fin 7573

This theorem is referenced by:  nnunifi  7822  oemapvali  8189  ttukeylem6  8944  limsucncmpi  31105