Theorem pw10b 4167

Description: The unit power class of a class is empty iff the class itself is empty. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
pw10b	⊢ (℘1A = ∅ ↔ A = ∅)

Proof of Theorem pw10b

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3560	. . . 4 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
2		snelpw1 4147	. . . . . 6 ⊢ ({x} ∈ ℘1A ↔ x ∈ A)
3		ne0i 3557	. . . . . 6 ⊢ ({x} ∈ ℘1A → ℘1A ≠ ∅)
4	2, 3	sylbir 204	. . . . 5 ⊢ (x ∈ A → ℘1A ≠ ∅)
5	4	exlimiv 1634	. . . 4 ⊢ (∃x x ∈ A → ℘1A ≠ ∅)
6	1, 5	sylbi 187	. . 3 ⊢ (A ≠ ∅ → ℘1A ≠ ∅)
7	6	necon4i 2577	. 2 ⊢ (℘1A = ∅ → A = ∅)
8		pw1eq 4144	. . 3 ⊢ (A = ∅ → ℘1A = ℘1∅)
9		pw10 4162	. . 3 ⊢ ℘1∅ = ∅
10	8, 9	syl6eq 2401	. 2 ⊢ (A = ∅ → ℘1A = ∅)
11	7, 10	impbii 180	1 ⊢ (℘1A = ∅ ↔ A = ∅)

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551  {csn 3738  ℘₁cpw1 4136

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  ax-nin 4079  ax-sn 4088

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-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is referenced by:  ncfinlower  4484