Theorem pw1sn 4165

Description: Compute the unit power class of a singleton. (Contributed by SF, 22-Jan-2015.)

Hypothesis

Ref	Expression
pw1sn.1	|- A e. _V

Assertion

Ref	Expression
pw1sn	|- ~P1 {A} = {{A}}

Proof of Theorem pw1sn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw1sn.1	. . . 4 |- A e. _V
2		sneq 3744	. . . . 5 |- (y = A -> {y} = {A})
3	2	eqeq2d 2364	. . . 4 |- (y = A -> (x = {y} <-> x = {A}))
4	1, 3	rexsn 3768	. . 3 |- (E.y e. {A}x = {y} <-> x = {A})
5		elpw1 4144	. . 3 |- (x e. ~P1 {A} <-> E.y e. {A}x = {y})
6		elsn 3748	. . 3 |- (x e. {{A}} <-> x = {A})
7	4, 5, 6	3bitr4i 268	. 2 |- (x e. ~P1 {A} <-> x e. {{A}})
8	7	eqriv 2350	1 |- ~P1 {A} = {{A}}

Syntax hints:   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859  {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-3an 936  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-rex 2620  df-v 2861  df-sbc 3047  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:  pw1eqadj  4332  ncfinraise  4481  tfinsuc  4498  sfindbl  4530  tc1c  6165  ce0nn  6180  ce2  6192