Theorem snelpwg 4114

Description: A singleton of a set belongs to a power class of a set containing it. (Contributed by SF, 1-Feb-2015.)

Assertion

Ref	Expression
snelpwg	⊢ (A ∈ V → ({A} ∈ ℘B ↔ A ∈ B))

Proof of Theorem snelpwg

Step	Hyp	Ref	Expression
1		snssg 3844	. 2 ⊢ (A ∈ V → (A ∈ B ↔ {A} ⊆ B))
2		snex 4111	. . 3 ⊢ {A} ∈ V
3	2	elpw 3728	. 2 ⊢ ({A} ∈ ℘B ↔ {A} ⊆ B)
4	1, 3	syl6rbbr 255	1 ⊢ (A ∈ V → ({A} ∈ ℘B ↔ A ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  {csn 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-pw 3724  df-sn 3741

This theorem is referenced by:  snelpw  4115