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Theorem opkabssvvk 4208
 Description: Any Kuratowski ordered pair abstraction is a subset of (V ×k V). (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
opkabssvvk {x yz(x = ⟪y, z φ)} (V ×k V)
Distinct variable groups:   x,y   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem opkabssvvk
Dummy variables w t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . . . . . . 7 y, z⟫ = ⟪y, z
2 vex 2862 . . . . . . . 8 y V
3 vex 2862 . . . . . . . 8 z V
4 opkeq12 4061 . . . . . . . . 9 ((w = y t = z) → ⟪w, t⟫ = ⟪y, z⟫)
54eqeq2d 2364 . . . . . . . 8 ((w = y t = z) → (⟪y, z⟫ = ⟪w, t⟫ ↔ ⟪y, z⟫ = ⟪y, z⟫))
62, 3, 5spc2ev 2947 . . . . . . 7 (⟪y, z⟫ = ⟪y, z⟫ → wty, z⟫ = ⟪w, t⟫)
71, 6ax-mp 5 . . . . . 6 wty, z⟫ = ⟪w, t
8 elvvk 4207 . . . . . 6 (⟪y, z (V ×k V) ↔ wty, z⟫ = ⟪w, t⟫)
97, 8mpbir 200 . . . . 5 y, z (V ×k V)
10 eleq1 2413 . . . . 5 (x = ⟪y, z⟫ → (x (V ×k V) ↔ ⟪y, z (V ×k V)))
119, 10mpbiri 224 . . . 4 (x = ⟪y, z⟫ → x (V ×k V))
1211adantr 451 . . 3 ((x = ⟪y, z φ) → x (V ×k V))
1312exlimivv 1635 . 2 (yz(x = ⟪y, z φ) → x (V ×k V))
1413abssi 3341 1 {x yz(x = ⟪y, z φ)} (V ×k V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859   ⊆ wss 3257  ⟪copk 4057   ×k cxpk 4174 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-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  df-xpk 4185 This theorem is referenced by:  opkabssvvki  4209
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