Theorem pw10b 4166

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	|- (~P1 A = (/) <-> A = (/))

Proof of Theorem pw10b

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3559	. . . 4 |- (A =/= (/) <-> E.x x e. A)
2		snelpw1 4146	. . . . . 6 |- ({x} e. ~P1 A <-> x e. A)
3		ne0i 3556	. . . . . 6 |- ({x} e. ~P1 A -> ~P1 A =/= (/))
4	2, 3	sylbir 204	. . . . 5 |- (x e. A -> ~P1 A =/= (/))
5	4	exlimiv 1634	. . . 4 |- (E.x x e. A -> ~P1 A =/= (/))
6	1, 5	sylbi 187	. . 3 |- (A =/= (/) -> ~P1 A =/= (/))
7	6	necon4i 2576	. 2 |- (~P1 A = (/) -> A = (/))
8		pw1eq 4143	. . 3 |- (A = (/) -> ~P1 A = ~P1 (/))
9		pw10 4161	. . 3 |- ~P1 (/) = (/)
10	8, 9	syl6eq 2401	. 2 |- (A = (/) -> ~P1 A = (/))
11	7, 10	impbii 180	1 |- (~P1 A = (/) <-> A = (/))

Syntax hints:   <-> wb 176  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2516  (/)c0 3550  {csn 3737  ~P₁ cpw1 4135

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137

This theorem is referenced by:  ncfinlower  4483