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Theorem ordunifi 7123
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 4591 . . . . . 6  |-  _E  We  On
2 weso 4400 . . . . . 6  |-  (  _E  We  On  ->  _E  Or  On )
31, 2ax-mp 8 . . . . 5  |-  _E  Or  On
4 soss 4348 . . . . 5  |-  ( A 
C_  On  ->  (  _E  Or  On  ->  _E  Or  A ) )
53, 4mpi 16 . . . 4  |-  ( A 
C_  On  ->  _E  Or  A )
6 fimax2g 7119 . . . 4  |-  ( (  _E  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  _E  y )
75, 6syl3an1 1215 . . 3  |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  _E  y )
8 ssel2 3188 . . . . . . . . 9  |-  ( ( A  C_  On  /\  y  e.  A )  ->  y  e.  On )
98adantlr 695 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  y  e.  A
)  ->  y  e.  On )
10 ssel2 3188 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
1110adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  y  e.  A
)  ->  x  e.  On )
12 ontri1 4442 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  C_  x  <->  -.  x  e.  y ) )
13 epel 4324 . . . . . . . . . 10  |-  ( x  _E  y  <->  x  e.  y )
1413notbii 287 . . . . . . . . 9  |-  ( -.  x  _E  y  <->  -.  x  e.  y )
1512, 14syl6rbbr 255 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( -.  x  _E  y  <->  y  C_  x
) )
169, 11, 15syl2anc 642 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  y  e.  A
)  ->  ( -.  x  _E  y  <->  y  C_  x ) )
1716ralbidva 2572 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  A. y  e.  A  y 
C_  x ) )
18 unissb 3873 . . . . . 6  |-  ( U. A  C_  x  <->  A. y  e.  A  y  C_  x )
1917, 18syl6bbr 254 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  U. A  C_  x )
)
2019rexbidva 2573 . . . 4  |-  ( A 
C_  On  ->  ( E. x  e.  A  A. y  e.  A  -.  x  _E  y  <->  E. x  e.  A  U. A  C_  x ) )
21203ad2ant1 976 . . 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 201 . 2  |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  U. A  C_  x )
23 elssuni 3871 . . . 4  |-  ( x  e.  A  ->  x  C_ 
U. A )
24 eqss 3207 . . . . 5  |-  ( x  =  U. A  <->  ( x  C_ 
U. A  /\  U. A  C_  x ) )
25 eleq1 2356 . . . . . 6  |-  ( x  =  U. A  -> 
( x  e.  A  <->  U. A  e.  A ) )
2625biimpcd 215 . . . . 5  |-  ( x  e.  A  ->  (
x  =  U. A  ->  U. A  e.  A
) )
2724, 26syl5bir 209 . . . 4  |-  ( x  e.  A  ->  (
( x  C_  U. A  /\  U. A  C_  x
)  ->  U. A  e.  A ) )
2823, 27mpand 656 . . 3  |-  ( x  e.  A  ->  ( U. A  C_  x  ->  U. A  e.  A
) )
2928rexlimiv 2674 . 2  |-  ( E. x  e.  A  U. A  C_  x  ->  U. A  e.  A )
3022, 29syl 15 1  |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  U. A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   U.cuni 3843   class class class wbr 4039    _E cep 4319    Or wor 4329    We wwe 4367   Oncon0 4408   Fincfn 6879
This theorem is referenced by:  nnunifi  7124  oemapvali  7402  ttukeylem6  8157  limsucncmpi  24956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-fin 6883
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