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 ) ) ) )