Theorem pwsn 4832

Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.)

Assertion

Ref	Expression
pwsn	⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssn 4787	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2	1	abbii 2830	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3		df-pw 4562	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
4		dfpr2 4605	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
5	2, 3, 4	3eqtr4i 2796	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570  {cab 2741   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591

This theorem is used by:  pmtrsn  19784  topsn  23304  conncompid  23804  lfuhgr1v0e  29892  esumsnf  34807  cvmlift2lem9  36284  mh-infprim2bi  37550  rrxtopn0b  47579  sge0sn  47662