Metamath Proof Explorer

Theorem ensymb

Description: Symmetry of equinumerosity. Theorem 2 of Suppes p. 92. (Contributed by Mario Carneiro, 26-Apr-2015)

Ref Expression
Assertion ensymb 
|- ( A ~~ B <-> B ~~ A )

Proof

Step Hyp Ref Expression
1 ener 
 |-  ~~ Er _V
2 1 a1i 
 |-  ( T. -> ~~ Er _V )
3 2 ersymb 
 |-  ( T. -> ( A ~~ B <-> B ~~ A ) )
4 3 mptru 
 |-  ( A ~~ B <-> B ~~ A )