Theorem snsspw 3804

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 3246	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
2		velsn 3649	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		df-pw 3617	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4	3	abeq2i 2315	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5	1, 2, 4	3imtr4i 201	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
6	5	ssriv 3196	1 ⊢ {𝐴} ⊆ 𝒫 𝐴

Syntax hints:   = wceq 1372   ∈ wcel 2175   ⊆ wss 3165  𝒫 cpw 3615  {csn 3632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638

This theorem is referenced by:  snexg  4227