Theorem pwsn 3882

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

Assertion

Ref	Expression
pwsn	⊢ ℘{A} = {∅, {A}}

Proof of Theorem pwsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssn 3865	. . 3 ⊢ (x ⊆ {A} ↔ (x = ∅ ∨ x = {A}))
2	1	abbii 2466	. 2 ⊢ {x ∣ x ⊆ {A}} = {x ∣ (x = ∅ ∨ x = {A})}
3		df-pw 3725	. 2 ⊢ ℘{A} = {x ∣ x ⊆ {A}}
4		dfpr2 3750	. 2 ⊢ {∅, {A}} = {x ∣ (x = ∅ ∨ x = {A})}
5	2, 3, 4	3eqtr4i 2383	1 ⊢ ℘{A} = {∅, {A}}

Syntax hints:   ∨ wo 357   = wceq 1642  {cab 2339   ⊆ wss 3258  ∅c0 3551  ℘cpw 3723  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)