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Theorem domen2 6513
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
Assertion
Ref Expression
domen2  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )

Proof of Theorem domen2
StepHypRef Expression
1 domentr 6462 . . 3  |-  ( ( C  ~<_  A  /\  A  ~~  B )  ->  C  ~<_  B )
21ancoms 264 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  A )  ->  C  ~<_  B )
3 ensym 6452 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 domentr 6462 . . . 4  |-  ( ( C  ~<_  B  /\  B  ~~  A )  ->  C  ~<_  A )
54ancoms 264 . . 3  |-  ( ( B  ~~  A  /\  C  ~<_  B )  ->  C  ~<_  A )
63, 5sylan 277 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  B )  ->  C  ~<_  A )
72, 6impbida 561 1  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   class class class wbr 3822    ~~ cen 6409    ~<_ cdom 6410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-er 6246  df-en 6412  df-dom 6413
This theorem is referenced by:  fihashdom  10111
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