New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  opkelopkab GIF version

Theorem opkelopkab 4246
 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 yz(x = ⟪y, z φ)}
opkelopkab.2 (y = B → (φψ))
opkelopkab.3 (z = C → (ψχ))
opkelopkab.4 B V
opkelopkab.5 C V
Assertion
Ref Expression
opkelopkab (⟪B, C Aχ)
Distinct variable groups:   y,A,z   x,B,y,z   x,C,y,z   χ,z   φ,x   ψ,y   x,y,z
Allowed substitution hints:   φ(y,z)   ψ(x,z)   χ(x,y)   A(x)

Proof of Theorem opkelopkab
StepHypRef Expression
1 opkelopkab.4 . 2 B V
2 opkelopkab.5 . 2 C V
3 opkelopkab.1 . . 3 A = {x yz(x = ⟪y, z φ)}
4 opkelopkab.2 . . 3 (y = B → (φψ))
5 opkelopkab.3 . . 3 (z = C → (ψχ))
63, 4, 5opkelopkabg 4245 . 2 ((B V C V) → (⟪B, C Aχ))
71, 2, 6mp2an 653 1 (⟪B, C Aχ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ⟪copk 4057 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 4078  ax-sn 4087 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by:  sikss1c1c  4267  dfima2  4745  dfco1  4748  dfsi2  4751
 Copyright terms: Public domain W3C validator