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Theorem onsdominel 7258
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 4617 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
21ancoms 441 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
3 inex1g 4348 . . . . . . 7  |-  ( A  e.  On  ->  ( A  i^i  C )  e. 
_V )
4 ssrin 3568 . . . . . . 7  |-  ( B 
C_  A  ->  ( B  i^i  C )  C_  ( A  i^i  C ) )
5 ssdomg 7155 . . . . . . 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 37 . . . . . 6  |-  ( A  e.  On  ->  ( B  C_  A  ->  ( B  i^i  C )  ~<_  ( A  i^i  C ) ) )
7 domnsym 7235 . . . . . 6  |-  ( ( B  i^i  C )  ~<_  ( A  i^i  C
)  ->  -.  ( A  i^i  C )  ~< 
( B  i^i  C
) )
86, 7syl6 32 . . . . 5  |-  ( A  e.  On  ->  ( B  C_  A  ->  -.  ( A  i^i  C ) 
~<  ( B  i^i  C
) ) )
98adantr 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  -.  ( A  i^i  C )  ~<  ( B  i^i  C ) ) )
102, 9sylbird 228 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  ->  -.  ( A  i^i  C )  ~< 
( B  i^i  C
) ) )
1110con4d 100 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  C )  ~<  ( B  i^i  C )  ->  A  e.  B ) )
12113impia 1151 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322   class class class wbr 4214   Oncon0 4583    ~<_ cdom 7109    ~< csdm 7110
This theorem is referenced by:  fin23lem27  8210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114
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