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Theorem wdomen1 8096
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
wdomen1  |-  ( A 
~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )

Proof of Theorem wdomen1
StepHypRef Expression
1 ensym 7623 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
2 endom 7601 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
3 domwdom 8094 . . . 4  |-  ( B  ~<_  A  ->  B  ~<_*  A )
41, 2, 33syl 18 . . 3  |-  ( A 
~~  B  ->  B  ~<_*  A )
5 wdomtr 8095 . . 3  |-  ( ( B  ~<_*  A  /\  A  ~<_*  C
)  ->  B  ~<_*  C )
64, 5sylan 474 . 2  |-  ( ( A  ~~  B  /\  A  ~<_*  C )  ->  B  ~<_*  C )
7 endom 7601 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
8 domwdom 8094 . . . 4  |-  ( A  ~<_  B  ->  A  ~<_*  B )
97, 8syl 17 . . 3  |-  ( A 
~~  B  ->  A  ~<_*  B )
10 wdomtr 8095 . . 3  |-  ( ( A  ~<_*  B  /\  B  ~<_*  C
)  ->  A  ~<_*  C )
119, 10sylan 474 . 2  |-  ( ( A  ~~  B  /\  B  ~<_*  C )  ->  A  ~<_*  C )
126, 11impbida 844 1  |-  ( A 
~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   class class class wbr 4405    ~~ cen 7571    ~<_ cdom 7572    ~<_* cwdom 8077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-wdom 8079
This theorem is referenced by: (None)
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