Theorem grothprimlem 8721

Description: Lemma for grothprim 8722. Expand the membership of an unordered pair into primitives.

Assertion

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

Distinct variable group:   w,v,u,h,g

Proof of Theorem grothprimlem

Step	Hyp	Ref	Expression
1		dfpr2 2418	. . 3 |- {u, v} = {h | (h = u \/ h = v)}
2	1	eleq1i 1534	. 2 |- ({u, v} e. w <-> {h | (h = u \/ h = v)} e. w)
3		clabel 1579	. 2 |- ({h | (h = u \/ h = v)} e. w <-> E.g(g e. w /\ A.h(h e. g <-> (h = u \/ h = v))))
4	2, 3	bitr 173	1 |- ({u, v} e. w <-> E.g(g e. w /\ A.h(h e. g <-> (h = u \/ h = v))))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  {cpr 2406

This theorem is referenced by:  grothprim 8722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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