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Metamath Proof Explorer

Theorem infpssALT

Description: Alternate proof of infpss , shorter but requiring Replacement ( ax-rep ). (Contributed by Stefan O'Rear, 30-Oct-2014) (Revised by Mario Carneiro, 16-May-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion infpssALT 
|- ( _om ~<_ A -> E. x ( x C. A /\ x ~~ A ) )

Proof

Step Hyp Ref Expression
1 ominf4 
 |-  -. _om e. Fin4
2 reldom 
 |-  Rel ~<_
3 2 brrelex2i 
 |-  ( _om ~<_ A -> A e. _V )
4 isfin4 
 |-  ( A e. _V -> ( A e. Fin4 <-> -. E. x ( x C. A /\ x ~~ A ) ) )
5 3 4 syl 
 |-  ( _om ~<_ A -> ( A e. Fin4 <-> -. E. x ( x C. A /\ x ~~ A ) ) )
6 domfin4 
 |-  ( ( A e. Fin4 /\ _om ~<_ A ) -> _om e. Fin4 )
7 6 expcom 
 |-  ( _om ~<_ A -> ( A e. Fin4 -> _om e. Fin4 ) )
8 5 7 sylbird 
 |-  ( _om ~<_ A -> ( -. E. x ( x C. A /\ x ~~ A ) -> _om e. Fin4 ) )
9 1 8 mt3i 
 |-  ( _om ~<_ A -> E. x ( x C. A /\ x ~~ A ) )