Theorem snelpw 2758

Description: A singleton of a set belongs to the power class of a class containing the set.

Hypothesis

Ref	Expression
snelpw.1	⊢ A ∈ V

Assertion

Ref	Expression
snelpw	⊢ (A ∈ B ↔ {A} ∈ ℘B)

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 ⊢ A ∈ V
2	1	snss 2465	. 2 ⊢ (A ∈ B ↔ {A} ⊆ B)
3		snex 2756	. . 3 ⊢ {A} ∈ V
4	3	elpw 2408	. 2 ⊢ ({A} ∈ ℘B ↔ {A} ⊆ B)
5	2, 4	bitr4 176	1 ⊢ (A ∈ B ↔ {A} ∈ ℘B)

Syntax hints:   ↔ wb 146   ∈ wcel 960  Vcvv 1814   ⊆ wss 2050  ℘cpw 2405  {csn 2413

This theorem is referenced by:  unipw 2762  canth2 4490  abfi 10443  dtt2 10589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416

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