Theorem pw0 4816

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 4406	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2806	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4606	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4631	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2772	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1536  {cab 2711   ⊆ wss 3962  ∅c0 4338  𝒫 cpw 4604  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-dif 3965  df-ss 3979  df-nul 4339  df-pw 4606  df-sn 4631

This theorem is referenced by:  p0ex  5389  pwfi  9354  ackbij1lem14  10269  fin1a2lem12  10448  0tsk  10792  hashbc  14488  incexclem  15868  sn0topon  23020  sn0cld  23113  ust0  24243  made0  27926  uhgr0vb  29103  uhgr0  29104  esumnul  34028  rankeq1o  36152  ssoninhaus  36430  sge00  46331