Theorem elvvk 4207

Description: Membership in (V ×_k V) (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
elvvk	⊢ (A ∈ (V ×k V) ↔ ∃x∃y A = ⟪x, y⟫)

Distinct variable group:   x,A,y

Proof of Theorem elvvk

Step	Hyp	Ref	Expression
1		elxpk 4196	. 2 ⊢ (A ∈ (V ×k V) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ V ∧ y ∈ V)))
2		vex 2862	. . . . 5 ⊢ x ∈ V
3		vex 2862	. . . . 5 ⊢ y ∈ V
4	2, 3	pm3.2i 441	. . . 4 ⊢ (x ∈ V ∧ y ∈ V)
5	4	biantru 491	. . 3 ⊢ (A = ⟪x, y⟫ ↔ (A = ⟪x, y⟫ ∧ (x ∈ V ∧ y ∈ V)))
6	5	2exbii 1583	. 2 ⊢ (∃x∃y A = ⟪x, y⟫ ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ V ∧ y ∈ V)))
7	1, 6	bitr4i 243	1 ⊢ (A ∈ (V ×k V) ↔ ∃x∃y A = ⟪x, y⟫)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟪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:  opkabssvvk  4208  ssrelk  4211  eqrelk  4212  insklem  4304  ltfinex  4464  setconslem4  4734  setconslem6  4736