Theorem pw0 4742

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 4328	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2809	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4532	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4559	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2776	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1539  {cab 2715   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559

This theorem is referenced by:  p0ex  5302  pwfi  8923  pwfiOLD  9044  ackbij1lem14  9920  fin1a2lem12  10098  0tsk  10442  hashbc  14093  incexclem  15476  sn0topon  22056  sn0cld  22149  ust0  23279  uhgr0vb  27345  uhgr0  27346  esumnul  31916  made0  33984  rankeq1o  34400  ssoninhaus  34564  sge00  43804