Theorem elssetkg 4270

Description: Membership via the Kuratowski subset relationship. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
elssetkg	⊢ ((A ∈ V ∧ B ∈ W) → (⟪{A}, B⟫ ∈ Sk ↔ A ∈ B))

Proof of Theorem elssetkg

Step	Hyp	Ref	Expression
1		snex 4112	. . 3 ⊢ {A} ∈ V
2		opkelssetkg 4269	. . 3 ⊢ (({A} ∈ V ∧ B ∈ W) → (⟪{A}, B⟫ ∈ Sk ↔ {A} ⊆ B))
3	1, 2	mpan 651	. 2 ⊢ (B ∈ W → (⟪{A}, B⟫ ∈ Sk ↔ {A} ⊆ B))
4		snssg 3845	. . 3 ⊢ (A ∈ V → (A ∈ B ↔ {A} ⊆ B))
5	4	bicomd 192	. 2 ⊢ (A ∈ V → ({A} ⊆ B ↔ A ∈ B))
6	3, 5	sylan9bbr 681	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪{A}, B⟫ ∈ Sk ↔ A ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  {csn 3738  ⟪copk 4058   S_k cssetk 4184

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-3an 936  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-ssetk 4194

This theorem is referenced by:  elssetk  4271  opkelimagekg  4272  setswith  4322