Theorem pw0 4745

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 4331	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2808	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4535	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4562	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2776	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1539  {cab 2715   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562

This theorem is referenced by:  p0ex  5307  pwfi  8961  pwfiOLD  9114  ackbij1lem14  9989  fin1a2lem12  10167  0tsk  10511  hashbc  14165  incexclem  15548  sn0topon  22148  sn0cld  22241  ust0  23371  uhgr0vb  27442  uhgr0  27443  esumnul  32016  made0  34057  rankeq1o  34473  ssoninhaus  34637  sge00  43914