Theorem eqrelkrdv 4215

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)
eqrelkrdv.3	⊢ (φ → (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B))

Assertion

Ref	Expression
eqrelkrdv	⊢ (φ → A = B)

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

Proof of Theorem eqrelkrdv

Step	Hyp	Ref	Expression
1		eqrelkrdv.3	. . 3 ⊢ (φ → (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B))
2	1	alrimivv 1632	. 2 ⊢ (φ → ∀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 4213	. . 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	sylibr 203	1 ⊢ (φ → A = B)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  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-ral 2620  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: (None)