Theorem snsspw 2483

Description: The singleton of a class is a subset of its power class.

Assertion

Ref	Expression
snsspw	⊢ {A} ⊆ ℘A

Proof of Theorem snsspw

Step	Hyp	Ref	Expression
1		eqimss 2112	. . 3 ⊢ (x = A → x ⊆ A)
2		elsn 2425	. . 3 ⊢ (x ∈ {A} ↔ x = A)
3		df-pw 2406	. . . 4 ⊢ ℘A = {x∣x ⊆ A}
4	3	abeq2i 1573	. . 3 ⊢ (x ∈ ℘A ↔ x ⊆ A)
5	1, 2, 4	3imtr4 219	. 2 ⊢ (x ∈ {A} → x ∈ ℘A)
6	5	ssriv 2072	1 ⊢ {A} ⊆ ℘A

Syntax hints:   = wceq 958   ∈ wcel 960   ⊆ wss 2050  ℘cpw 2405  {csn 2413

This theorem is referenced by:  snex 2756

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-pw 2406  df-sn 2416

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