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 4761	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2	1	abbii 2888	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3		df-pw 4543	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
4		dfpr2 4588	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
5	2, 3, 4	3eqtr4i 2856	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Syntax hints:   ∨ wo 843   = wceq 1537  {cab 2801   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {csn 4569  {cpr 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-pw 4543  df-sn 4570  df-pr 4572

This theorem is referenced by:  pmtrsn  18649  topsn  21541  conncompid  22041  lfuhgr1v0e  27038  esumsnf  31325  cvmlift2lem9  32560  rrxtopn0b  42588  sge0sn  42668