Theorem pwsn 4860

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 2828	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3		df-pw 4559	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
4		dfpr2 4605	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
5	2, 3, 4	3eqtr4i 2794	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570  {cab 2739   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  {cpr 4586

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-pw 4559  df-sn 4585  df-pr 4587

This theorem is used by:  pmtrsn  19726  topsn  23242  conncompid  23742  lfuhgr1v0e  29828  esumsnf  34689  cvmlift2lem9  36055  mh-infprim2bi  37315  rrxtopn0b  47275  sge0sn  47358