Theorem opkelopkab 4247

Description: Kuratowski ordered pair membership in an abstraction of Kuratowski ordered pairs. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
opkelopkab.1	|- A = {x | E.yE.z(x = <<y, z>> /\ ph)}
opkelopkab.2	|- (y = B -> (ph <-> ps))
opkelopkab.3	|- (z = C -> (ps <-> ch))
opkelopkab.4	|- B e. _V
opkelopkab.5	|- C e. _V

Assertion

Ref	Expression
opkelopkab	|- (<<B, C>> e. A <-> ch)

Distinct variable groups:   y,A,z   x,B,y,z   x,C,y,z   ch,z   ph,x   ps,y   x,y,z

Allowed substitution hints:   ph(y,z)   ps(x,z)   ch(x,y)   A(x)

Proof of Theorem opkelopkab

Step	Hyp	Ref	Expression
1		opkelopkab.4	. 2 |- B e. _V
2		opkelopkab.5	. 2 |- C e. _V
3		opkelopkab.1	. . 3 |- A = {x | E.yE.z(x = <<y, z>> /\ ph)}
4		opkelopkab.2	. . 3 |- (y = B -> (ph <-> ps))
5		opkelopkab.3	. . 3 |- (z = C -> (ps <-> ch))
6	3, 4, 5	opkelopkabg 4246	. 2 |- ((B e. _V /\ C e. _V) -> (<<B, C>> e. A <-> ch))
7	1, 2, 6	mp2an 653	1 |- (<<B, C>> e. A <-> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860  <<copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  sikss1c1c  4268  dfima2  4746  dfco1  4749  dfsi2  4752