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Theorem wdomen1 7789
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 7356 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
2 endom 7334 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
3 domwdom 7787 . . . 4  |-  ( B  ~<_  A  ->  B  ~<_*  A )
41, 2, 33syl 20 . . 3  |-  ( A 
~~  B  ->  B  ~<_*  A )
5 wdomtr 7788 . . 3  |-  ( ( B  ~<_*  A  /\  A  ~<_*  C
)  ->  B  ~<_*  C )
64, 5sylan 471 . 2  |-  ( ( A  ~~  B  /\  A  ~<_*  C )  ->  B  ~<_*  C )
7 endom 7334 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
8 domwdom 7787 . . . 4  |-  ( A  ~<_  B  ->  A  ~<_*  B )
97, 8syl 16 . . 3  |-  ( A 
~~  B  ->  A  ~<_*  B )
10 wdomtr 7788 . . 3  |-  ( ( A  ~<_*  B  /\  B  ~<_*  C
)  ->  A  ~<_*  C )
119, 10sylan 471 . 2  |-  ( ( A  ~~  B  /\  B  ~<_*  C )  ->  A  ~<_*  C )
126, 11impbida 828 1  |-  ( A 
~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4290    ~~ cen 7305    ~<_ cdom 7306    ~<_* cwdom 7770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-wdom 7772
This theorem is referenced by: (None)
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