Theorem pw0 4161

Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) (Contributed by SF, 5-Aug-1993.) (Revised by SF, 29-Jun-2011.)

Assertion

Ref	Expression
pw0	⊢ ℘∅ = {∅}

Proof of Theorem pw0

Step	Hyp	Ref	Expression
1		ss0b 3581	. . 3 ⊢ (x ⊆ ∅ ↔ x = ∅)
2	1	abbii 2466	. 2 ⊢ {x ∣ x ⊆ ∅} = {x ∣ x = ∅}
3		df-pw 3725	. 2 ⊢ ℘∅ = {x ∣ x ⊆ ∅}
4		df-sn 3742	. 2 ⊢ {∅} = {x ∣ x = ∅}
5	2, 3, 4	3eqtr4i 2383	1 ⊢ ℘∅ = {∅}

Syntax hints:   = wceq 1642  {cab 2339   ⊆ wss 3258  ∅c0 3551  ℘cpw 3723  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742

This theorem is referenced by:  pw10  4162  nnpweq  4524  sfin01  4529