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Theorem onsdominel 7010
Description: An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
onsdominel  |-  ( ( A  e.  On  /\  B  e.  On  /\  ( A  i^i  C )  ~< 
( B  i^i  C
) )  ->  A  e.  B )

Proof of Theorem onsdominel
StepHypRef Expression
1 ontri1 4426 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
21ancoms 439 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
3 inex1g 4157 . . . . . . 7  |-  ( A  e.  On  ->  ( A  i^i  C )  e. 
_V )
4 ssrin 3394 . . . . . . 7  |-  ( B 
C_  A  ->  ( B  i^i  C )  C_  ( A  i^i  C ) )
5 ssdomg 6907 . . . . . . 7  |-  ( ( A  i^i  C )  e.  _V  ->  (
( B  i^i  C
)  C_  ( A  i^i  C )  ->  ( B  i^i  C )  ~<_  ( A  i^i  C ) ) )
63, 4, 5syl2im 34 . . . . . 6  |-  ( A  e.  On  ->  ( B  C_  A  ->  ( B  i^i  C )  ~<_  ( A  i^i  C ) ) )
7 domnsym 6987 . . . . . 6  |-  ( ( B  i^i  C )  ~<_  ( A  i^i  C
)  ->  -.  ( A  i^i  C )  ~< 
( B  i^i  C
) )
86, 7syl6 29 . . . . 5  |-  ( A  e.  On  ->  ( B  C_  A  ->  -.  ( A  i^i  C ) 
~<  ( B  i^i  C
) ) )
98adantr 451 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  -.  ( A  i^i  C )  ~<  ( B  i^i  C ) ) )
102, 9sylbird 226 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  ->  -.  ( A  i^i  C )  ~< 
( B  i^i  C
) ) )
1110con4d 97 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  C )  ~<  ( B  i^i  C )  ->  A  e.  B ) )
12113impia 1148 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  ( A  i^i  C )  ~< 
( B  i^i  C
) )  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023   Oncon0 4392    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  fin23lem27  7954
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-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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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