Theorem snsspw 3526

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 2991	. . 3 ⊢ (x = A → x ⊆ A)
2		elsn 3382	. . 3 ⊢ (x ∈ {A} ↔ x = A)
3		df-pw 3353	. . . 4 ⊢ 𝒫 A = {x ∣ x ⊆ A}
4	3	abeq2i 2145	. . 3 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A)
5	1, 2, 4	3imtr4i 190	. 2 ⊢ (x ∈ {A} → x ∈ 𝒫 A)
6	5	ssriv 2943	1 ⊢ {A} ⊆ 𝒫 A

Syntax hints:   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351  {csn 3367

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  snexgOLD  3926  snexg  3927