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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.
StepHypRef Expression
1 epweon 6610 . . . . . 6  |-  _E  We  On
2 weso 4825 . . . . . 6  |-  (  _E  We  On  ->  _E  Or  On )
31, 2ax-mp 5 . . . . 5  |-  _E  Or  On
4 soss 4773 . . . . 5  |-  ( A 
C_  On  ->  (  _E  Or  On  ->  _E  Or  A ) )
53, 4mpi 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 )
75, 6syl3an1 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 )
98adantlr 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 )
1110adantr 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 )
1413notbii 298 . . . . . . . . 9  |-  ( -.  x  _E  y  <->  -.  x  e.  y )
1512, 14syl6rbbr 268 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( -.  x  _E  y  <->  y  C_  x
) )
169, 11, 15syl2anc 667 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  y  e.  A
)  ->  ( -.  x  _E  y  <->  y  C_  x ) )
1716ralbidva 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 )
1917, 18syl6bbr 267 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  U. A  C_  x )
)
2019rexbidva 2898 . . . 4  |-  ( A 
C_  On  ->  ( E. x  e.  A  A. y  e.  A  -.  x  _E  y  <->  E. x  e.  A  U. A  C_  x ) )
21203ad2ant1 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 ) )
227, 21mpbid 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 ) )
2625biimpcd 228 . . . . 5  |-  ( x  e.  A  ->  (
x  =  U. A  ->  U. A  e.  A
) )
2724, 26syl5bir 222 . . . 4  |-  ( x  e.  A  ->  (
( x  C_  U. A  /\  U. A  C_  x
)  ->  U. A  e.  A ) )
2823, 27mpand 681 . . 3  |-  ( x  e.  A  ->  ( U. A  C_  x  ->  U. A  e.  A
) )
2928rexlimiv 2873 . 2  |-  ( E. x  e.  A  U. A  C_  x  ->  U. A  e.  A )
3022, 29syl 17 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 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
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