This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem en2eqpr

Description: Building a set with two elements. (Contributed by FL, 11-Aug-2008) (Revised by Mario Carneiro, 10-Sep-2015)

Ref Expression
Assertion en2eqpr 
|- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )

Proof

Step Hyp Ref Expression
1 2onn 
 |-  2o e. _om
2 nnfi 
 |-  ( 2o e. _om -> 2o e. Fin )
3 1 2 ax-mp 
 |-  2o e. Fin
4 simpl1 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C ~~ 2o )
5 enfii 
 |-  ( ( 2o e. Fin /\ C ~~ 2o ) -> C e. Fin )
6 3 4 5 sylancr 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C e. Fin )
7 simpl2 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> A e. C )
8 simpl3 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> B e. C )
9 7 8 prssd 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } C_ C )
10 enpr2 
 |-  ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } ~~ 2o )
11 10 3expa 
 |-  ( ( ( A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ 2o )
12 11 3adantl1 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ 2o )
13 4 ensymd 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> 2o ~~ C )
14 entr 
 |-  ( ( { A , B } ~~ 2o /\ 2o ~~ C ) -> { A , B } ~~ C )
15 12 13 14 syl2anc 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ C )
16 fisseneq 
 |-  ( ( C e. Fin /\ { A , B } C_ C /\ { A , B } ~~ C ) -> { A , B } = C )
17 6 9 15 16 syl3anc 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } = C )
18 17 eqcomd 
 |-  ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C = { A , B } )
19 18 ex 
 |-  ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )