Theorem grothprimlem 10140

Description: Lemma for grothprim 10141. 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 3059	. . 3 |- {u, v} = {h | (h = u \/ h = v)}
2	1	eleq1i 1960	. 2 |- ({u, v} e. w <-> {h | (h = u \/ h = v)} e. w)
3		clabel 2014	. 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	bitri 190	1 |- ({u, v} e. w <-> E.g(g e. w /\ A.h(h e. g <-> (h = u \/ h = v))))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  {cpr 3045

This theorem is referenced by:  grothprim 10141

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050

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