Theorem pw1sn 4166

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

Hypothesis

Ref	Expression
pw1sn.1	⊢ A ∈ V

Assertion

Ref	Expression
pw1sn	⊢ ℘1{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 ∈ V
2		sneq 3745	. . . . 5 ⊢ (y = A → {y} = {A})
3	2	eqeq2d 2364	. . . 4 ⊢ (y = A → (x = {y} ↔ x = {A}))
4	1, 3	rexsn 3769	. . 3 ⊢ (∃y ∈ {A}x = {y} ↔ x = {A})
5		elpw1 4145	. . 3 ⊢ (x ∈ ℘1{A} ↔ ∃y ∈ {A}x = {y})
6		elsn 3749	. . 3 ⊢ (x ∈ {{A}} ↔ x = {A})
7	4, 5, 6	3bitr4i 268	. 2 ⊢ (x ∈ ℘1{A} ↔ x ∈ {{A}})
8	7	eqriv 2350	1 ⊢ ℘1{A} = {{A}}

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  {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-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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-sbc 3048  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:  pw1eqadj  4333  ncfinraise  4482  tfinsuc  4499  sfindbl  4531  tc1c  6166  ce0nn  6181  ce2  6193