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Theorem onsdominel 7026
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 4442 . . . . 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 4173 . . . . . . 7  |-  ( A  e.  On  ->  ( A  i^i  C )  e. 
_V )
4 ssrin 3407 . . . . . . 7  |-  ( B 
C_  A  ->  ( B  i^i  C )  C_  ( A  i^i  C ) )
5 ssdomg 6923 . . . . . . 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 7003 . . . . . 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 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   class class class wbr 4039   Oncon0 4408    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  fin23lem27  7970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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