Theorem opkabssvvki 4210

Description: Any Kuratowski ordered pair abstraction is a subset of (V ×_k V). (Contributed by SF, 13-Jan-2015.)

Hypothesis

Ref	Expression
opkabssvvki.1	⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}

Assertion

Ref	Expression
opkabssvvki	⊢ A ⊆ (V ×k V)

Distinct variable groups:   x,y   x,z

Allowed substitution hints:   φ(x,y,z)   A(x,y,z)

Proof of Theorem opkabssvvki

Step	Hyp	Ref	Expression
1		opkabssvvki.1	. 2 ⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}
2		opkabssvvk 4209	. 2 ⊢ {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)} ⊆ (V ×k V)
3	1, 2	eqsstri 3302	1 ⊢ A ⊆ (V ×k V)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  Vcvv 2860   ⊆ wss 3258  ⟪copk 4058   ×_k cxpk 4175

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-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  df-xpk 4186

This theorem is referenced by:  xpkssvvk  4211  sikssvvk  4267  cnvkssvvk  4276  ssetkssvvk  4279  ins2kss  4280  ins3kss  4281  idkssvvk  4282