Theorem pwsn 4844

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 4773	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2	1	abbii 2806	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3		df-pw 4549	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
4		dfpr2 4592	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
5	2, 3, 4	3eqtr4i 2774	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Syntax hints:   ∨ wo 844   = wceq 1540  {cab 2713   ⊆ wss 3898  ∅c0 4269  𝒫 cpw 4547  {csn 4573  {cpr 4575

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-pw 4549  df-sn 4574  df-pr 4576

This theorem is referenced by:  pmtrsn  19223  topsn  22186  conncompid  22688  lfuhgr1v0e  27910  esumsnf  32330  cvmlift2lem9  33572  rrxtopn0b  44181  sge0sn  44262