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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.
StepHypRef Expression
1 epweon 6400 . . . . . 6  |-  _E  We  On
2 weso 4716 . . . . . 6  |-  (  _E  We  On  ->  _E  Or  On )
31, 2ax-mp 5 . . . . 5  |-  _E  Or  On
4 soss 4664 . . . . 5  |-  ( A 
C_  On  ->  (  _E  Or  On  ->  _E  Or  A ) )
53, 4mpi 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 )
75, 6syl3an1 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 )
98adantlr 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 )
1110adantr 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 )
1413notbii 296 . . . . . . . . 9  |-  ( -.  x  _E  y  <->  -.  x  e.  y )
1512, 14syl6rbbr 264 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( -.  x  _E  y  <->  y  C_  x
) )
169, 11, 15syl2anc 661 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  y  e.  A
)  ->  ( -.  x  _E  y  <->  y  C_  x ) )
1716ralbidva 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 )
1917, 18syl6bbr 263 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  U. A  C_  x )
)
2019rexbidva 2737 . . . 4  |-  ( A 
C_  On  ->  ( E. x  e.  A  A. y  e.  A  -.  x  _E  y  <->  E. x  e.  A  U. A  C_  x ) )
21203ad2ant1 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 ) )
227, 21mpbid 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 ) )
2625biimpcd 224 . . . . 5  |-  ( x  e.  A  ->  (
x  =  U. A  ->  U. A  e.  A
) )
2724, 26syl5bir 218 . . . 4  |-  ( x  e.  A  ->  (
( x  C_  U. A  /\  U. A  C_  x
)  ->  U. A  e.  A ) )
2823, 27mpand 675 . . 3  |-  ( x  e.  A  ->  ( U. A  C_  x  ->  U. A  e.  A
) )
2928rexlimiv 2840 . 2  |-  ( E. x  e.  A  U. A  C_  x  ->  U. A  e.  A )
3022, 29syl 16 1  |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  U. A  e.  A )
Colors of variables: wff setvar class
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
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