Theorem pw0 4658

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 4277	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2863	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4461	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4479	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2831	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1525  {cab 2777   ⊆ wss 3865  ∅c0 4217  𝒫 cpw 4459  {csn 4478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-dif 3868  df-in 3872  df-ss 3880  df-nul 4218  df-pw 4461  df-sn 4479

This theorem is referenced by:  p0ex  5182  pwfi  8672  ackbij1lem14  9508  fin1a2lem12  9686  0tsk  10030  hashbc  13663  incexclem  15028  sn0topon  21294  sn0cld  21386  ust0  22515  uhgr0vb  26544  uhgr0  26545  esumnul  30920  rankeq1o  33243  ssoninhaus  33407  sge00  42222