Theorem pw0 4776

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 4364	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2796	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4565	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4590	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2762	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1540  {cab 2707   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-dif 3917  df-ss 3931  df-nul 4297  df-pw 4565  df-sn 4590

This theorem is referenced by:  p0ex  5339  pwfi  9268  ackbij1lem14  10185  fin1a2lem12  10364  0tsk  10708  hashbc  14418  incexclem  15802  sn0topon  22885  sn0cld  22977  ust0  24107  made0  27785  uhgr0vb  28999  uhgr0  29000  esumnul  34038  rankeq1o  36159  ssoninhaus  36436  sge00  46374