Theorem pw0 4833

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 4354	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2830	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4562	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4589	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2796	1 ⊢ 𝒫 ∅ = {∅}

This proof depends on syntax axioms:   = wceq 1570  {cab 2741   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-dif 4050  df-nul 4282  df-pw 4562  df-sn 4589

This theorem is used by:  p0ex  5351  pwfi  9341  ackbij1lem14  10340  fin1a2lem12  10519  0tsk  10870  hashbc  14628  incexclem  16037  sn0topon  23349  sn0cld  23441  ust0  24572  made0  28282  uhgr0vb  29683  uhgr0  29684  vieta  34295  esumnul  34763  r11  35822  rankeq1o  37123  ssoninhaus  37427  sge00  47601