Theorem eqrelkriiv 4213

Description: Equality for two Kuratowski relationships. (Contributed by SF, 13-Jan-2015.)

Hypotheses

Ref	Expression
eqrelkriiv.1	⊢ A ⊆ (V ×k V)
eqrelkriiv.2	⊢ B ⊆ (V ×k V)
eqrelkriiv.3	⊢ (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B)

Assertion

Ref	Expression
eqrelkriiv	⊢ A = B

Distinct variable groups:   x,A,y   x,B,y

Proof of Theorem eqrelkriiv

Step	Hyp	Ref	Expression
1		eqrelkriiv.3	. . 3 ⊢ (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B)
2	1	gen2 1547	. 2 ⊢ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B)
3		eqrelkriiv.1	. . 3 ⊢ A ⊆ (V ×k V)
4		eqrelkriiv.2	. . 3 ⊢ B ⊆ (V ×k V)
5		eqrelk 4212	. . 3 ⊢ ((A ⊆ (V ×k V) ∧ B ⊆ (V ×k V)) → (A = B ↔ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B)))
6	3, 4, 5	mp2an 653	. 2 ⊢ (A = B ↔ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B))
7	2, 6	mpbir 200	1 ⊢ A = B

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  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-ral 2619  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:  cnvkxpk  4276  inxpk  4277  dfidk2  4313