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 ) ) ) ) |
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 ) ) ) ) |