Metamath Proof Explorer


Theorem grothprimlem

Description: Lemma for grothprim . Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007)

Ref Expression
Assertion grothprimlem
|- ( { u , v } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )

Proof

Step Hyp Ref Expression
1 dfpr2
 |-  { u , v } = { h | ( h = u \/ h = v ) }
2 1 eleq1i
 |-  ( { u , v } e. w <-> { h | ( h = u \/ h = v ) } e. w )
3 clabel
 |-  ( { h | ( h = u \/ h = v ) } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )
4 2 3 bitri
 |-  ( { u , v } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )