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Theorem wdomen1 7991
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 7554 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
2 endom 7532 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
3 domwdom 7989 . . . 4  |-  ( B  ~<_  A  ->  B  ~<_*  A )
41, 2, 33syl 20 . . 3  |-  ( A 
~~  B  ->  B  ~<_*  A )
5 wdomtr 7990 . . 3  |-  ( ( B  ~<_*  A  /\  A  ~<_*  C
)  ->  B  ~<_*  C )
64, 5sylan 471 . 2  |-  ( ( A  ~~  B  /\  A  ~<_*  C )  ->  B  ~<_*  C )
7 endom 7532 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
8 domwdom 7989 . . . 4  |-  ( A  ~<_  B  ->  A  ~<_*  B )
97, 8syl 16 . . 3  |-  ( A 
~~  B  ->  A  ~<_*  B )
10 wdomtr 7990 . . 3  |-  ( ( A  ~<_*  B  /\  B  ~<_*  C
)  ->  A  ~<_*  C )
119, 10sylan 471 . 2  |-  ( ( A  ~~  B  /\  B  ~<_*  C )  ->  A  ~<_*  C )
126, 11impbida 829 1  |-  ( A 
~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4440    ~~ cen 7503    ~<_ cdom 7504    ~<_* cwdom 7972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-wdom 7974
This theorem is referenced by: (None)
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