Theorem pw0 4763

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 4350	. . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
2	1	abbii 2800	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅}
3		df-pw 4551	. 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅}
4		df-sn 4576	. 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
5	2, 3, 4	3eqtr4i 2766	1 ⊢ 𝒫 ∅ = {∅}

Syntax hints:   = wceq 1541  {cab 2711   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-dif 3901  df-ss 3915  df-nul 4283  df-pw 4551  df-sn 4576

This theorem is referenced by:  p0ex  5324  pwfi  9210  ackbij1lem14  10130  fin1a2lem12  10309  0tsk  10653  hashbc  14362  incexclem  15745  sn0topon  22914  sn0cld  23006  ust0  24136  made0  27819  uhgr0vb  29052  uhgr0  29053  esumnul  34082  r11  35126  rankeq1o  36236  ssoninhaus  36513  sge00  46498