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Metamath Proof Explorer

Theorem wdomen2

Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015)

Ref Expression
Assertion wdomen2 
|- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )

Proof

Step Hyp Ref Expression
1 id 
 |-  ( C ~<_* A -> C ~<_* A )
2 endom 
 |-  ( A ~~ B -> A ~<_ B )
3 domwdom 
 |-  ( A ~<_ B -> A ~<_* B )
4 2 3 syl 
 |-  ( A ~~ B -> A ~<_* B )
5 wdomtr 
 |-  ( ( C ~<_* A /\ A ~<_* B ) -> C ~<_* B )
6 1 4 5 syl2anr 
 |-  ( ( A ~~ B /\ C ~<_* A ) -> C ~<_* B )
7 id 
 |-  ( C ~<_* B -> C ~<_* B )
8 ensym 
 |-  ( A ~~ B -> B ~~ A )
9 endom 
 |-  ( B ~~ A -> B ~<_ A )
10 domwdom 
 |-  ( B ~<_ A -> B ~<_* A )
11 8 9 10 3syl 
 |-  ( A ~~ B -> B ~<_* A )
12 wdomtr 
 |-  ( ( C ~<_* B /\ B ~<_* A ) -> C ~<_* A )
13 7 11 12 syl2anr 
 |-  ( ( A ~~ B /\ C ~<_* B ) -> C ~<_* A )
14 6 13 impbida 
 |-  ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )