Theorem grothprimlem 8789

Description: Lemma for grothprim 8790. 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 2432	. . 3 |- {u, v} = {h | (h = u \/ h = v)}
2	1	eleq1i 1544	. 2 |- ({u, v} e. w <-> {h | (h = u \/ h = v)} e. w)
3		clabel 1589	. 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 958   = wceq 960   e. wcel 962  E.wex 984  {cab 1470  {cpr 2420

This theorem is referenced by:  grothprim 8790

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-v 1819  df-un 2059  df-sn 2422  df-pr 2423

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