Theorem pwsn 4843

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 4769	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2	1	abbii 2803	. 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 2769	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Syntax hints:   ∨ wo 848   = wceq 1542  {cab 2714   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-pw 4543  df-sn 4568  df-pr 4570

This theorem is referenced by:  pmtrsn  19494  topsn  22896  conncompid  23396  lfuhgr1v0e  29323  esumsnf  34208  cvmlift2lem9  35493  mh-infprim2bi  36729  rrxtopn0b  46724  sge0sn  46807