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Theorem ordunifi 7788
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 6618 . . . . . 6  |-  _E  We  On
2 weso 4879 . . . . . 6  |-  (  _E  We  On  ->  _E  Or  On )
31, 2ax-mp 5 . . . . 5  |-  _E  Or  On
4 soss 4827 . . . . 5  |-  ( A 
C_  On  ->  (  _E  Or  On  ->  _E  Or  A ) )
53, 4mpi 17 . . . 4  |-  ( A 
C_  On  ->  _E  Or  A )
6 fimax2g 7784 . . . 4  |-  ( (  _E  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  _E  y )
75, 6syl3an1 1261 . . 3  |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  _E  y )
8 ssel2 3494 . . . . . . . . 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 3494 . . . . . . . . 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 4921 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( y  C_  x  <->  -.  x  e.  y ) )
13 epel 4803 . . . . . . . . . 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 2893 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A. y  e.  A  -.  x  _E  y  <->  A. y  e.  A  y 
C_  x ) )
18 unissb 4283 . . . . . 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 2965 . . . 4  |-  ( A 
C_  On  ->  ( E. x  e.  A  A. y  e.  A  -.  x  _E  y  <->  E. x  e.  A  U. A  C_  x ) )
21203ad2ant1 1017 . . 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 4281 . . . 4  |-  ( x  e.  A  ->  x  C_ 
U. A )
24 eqss 3514 . . . . 5  |-  ( x  =  U. A  <->  ( x  C_ 
U. A  /\  U. A  C_  x ) )
25 eleq1 2529 . . . . . 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 2943 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   U.cuni 4251   class class class wbr 4456    _E cep 4798    Or wor 4808    We wwe 4846   Oncon0 4887   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-er 7329  df-en 7536  df-fin 7539
This theorem is referenced by:  nnunifi  7789  oemapvali  8120  ttukeylem6  8911  limsucncmpi  30115
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