Theorem pw0 4812

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 4401	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2809	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4602	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4627	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2775	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1540  {cab 2714   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627

This theorem is referenced by:  p0ex  5384  pwfi  9357  ackbij1lem14  10272  fin1a2lem12  10451  0tsk  10795  hashbc  14492  incexclem  15872  sn0topon  23005  sn0cld  23098  ust0  24228  made0  27912  uhgr0vb  29089  uhgr0  29090  esumnul  34049  rankeq1o  36172  ssoninhaus  36449  sge00  46391