Theorem snsspw 3509

Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
snsspw	⊢ {A} ⊆ 𝒫 A

Proof of Theorem snsspw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqimss 2974	. . 3 ⊢ (x = A → x ⊆ A)
2		elsn 3365	. . 3 ⊢ (x ∈ {A} ↔ x = A)
3		df-pw 3336	. . . 4 ⊢ 𝒫 A = {x ∣ x ⊆ A}
4	3	abeq2i 2130	. . 3 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A)
5	1, 2, 4	3imtr4i 190	. 2 ⊢ (x ∈ {A} → x ∈ 𝒫 A)
6	5	ssriv 2926	1 ⊢ {A} ⊆ 𝒫 A

Syntax hints:   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894  𝒫 cpw 3334  {csn 3350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356

This theorem is referenced by:  snexgOLD  3909  snexg  3910