Theorem pw0 4779

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 4367	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2797	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4568	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4593	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2763	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1540  {cab 2708   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-dif 3920  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593

This theorem is referenced by:  p0ex  5342  pwfi  9275  ackbij1lem14  10192  fin1a2lem12  10371  0tsk  10715  hashbc  14425  incexclem  15809  sn0topon  22892  sn0cld  22984  ust0  24114  made0  27792  uhgr0vb  29006  uhgr0  29007  esumnul  34045  rankeq1o  36166  ssoninhaus  36443  sge00  46381