Theorem pw0 4768

Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
pw0	⊢ 𝒫 ∅ = {∅}

Proof of Theorem pw0

Step	Hyp	Ref	Expression
1		ss0b 4353	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2803	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4556	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4581	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2769	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1541  {cab 2714   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-dif 3904  df-ss 3918  df-nul 4286  df-pw 4556  df-sn 4581

This theorem is referenced by:  p0ex  5329  pwfi  9219  ackbij1lem14  10142  fin1a2lem12  10321  0tsk  10666  hashbc  14376  incexclem  15759  sn0topon  22942  sn0cld  23034  ust0  24164  made0  27859  uhgr0vb  29145  uhgr0  29146  vieta  33736  esumnul  34205  r11  35250  rankeq1o  36365  ssoninhaus  36642  sge00  46616