Theorem pw0 4821

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 4402	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2796	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4609	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4634	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2764	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1534  {cab 2703   ⊆ wss 3947  ∅c0 4325  𝒫 cpw 4607  {csn 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-dif 3950  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634

This theorem is referenced by:  p0ex  5388  pwfi  9359  pwfiOLD  9391  ackbij1lem14  10276  fin1a2lem12  10454  0tsk  10798  hashbc  14470  incexclem  15840  sn0topon  22992  sn0cld  23085  ust0  24215  made0  27897  uhgr0vb  29008  uhgr0  29009  esumnul  33881  rankeq1o  35995  ssoninhaus  36160  sge00  45997