Theorem snelpw 1861

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 1849	. 2 ⊢ (A ∈ B ↔ {A} ⊆ B)
3		snex 1859	. . 3 ⊢ {A} ∈ V
4	3	elpw 1801	. 2 ⊢ ({A} ∈ ℘B ↔ {A} ⊆ B)
5	2, 4	bitr4 154	1 ⊢ (A ∈ B ↔ {A} ∈ ℘B)

Syntax hints:   ↔ wb 127   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  {csn 1808

This theorem is referenced by:  unipw 1960  canth2 3381

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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