Theorem pw0 4788

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 4376	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2802	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4577	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4602	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2768	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1540  {cab 2713   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {csn 4601

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 2707

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 2714  df-cleq 2727  df-clel 2809  df-dif 3929  df-ss 3943  df-nul 4309  df-pw 4577  df-sn 4602

This theorem is referenced by:  p0ex  5354  pwfi  9329  ackbij1lem14  10246  fin1a2lem12  10425  0tsk  10769  hashbc  14471  incexclem  15852  sn0topon  22936  sn0cld  23028  ust0  24158  made0  27837  uhgr0vb  29051  uhgr0  29052  esumnul  34079  rankeq1o  36189  ssoninhaus  36466  sge00  46405