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Theorem ordunifi 7107
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 4575 . . . . . 6  |-  _E  We  On
2 weso 4384 . . . . . 6  |-  (  _E  We  On  ->  _E  Or  On )
31, 2ax-mp 8 . . . . 5  |-  _E  Or  On
4 soss 4332 . . . . 5  |-  ( A 
C_  On  ->  (  _E  Or  On  ->  _E  Or  A ) )
53, 4mpi 16 . . . 4  |-  ( A 
C_  On  ->  _E  Or  A )
6 fimax2g 7103 . . . 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 3175 . . . . . . . . 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 3175 . . . . . . . . 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 4426 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  C_  x  <->  -.  x  e.  y ) )
13 epel 4308 . . . . . . . . . 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 2559 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  A. y  e.  A  y 
C_  x ) )
18 unissb 3857 . . . . . 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 2560 . . . 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 3855 . . . 4  |-  ( x  e.  A  ->  x  C_ 
U. A )
24 eqss 3194 . . . . 5  |-  ( x  =  U. A  <->  ( x  C_ 
U. A  /\  U. A  C_  x ) )
25 eleq1 2343 . . . . . 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 2661 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   U.cuni 3827   class class class wbr 4023    _E cep 4303    Or wor 4313    We wwe 4351   Oncon0 4392   Fincfn 6863
This theorem is referenced by:  nnunifi  7108  oemapvali  7386  ttukeylem6  8141  limsucncmpi  24884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-fin 6867
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