Theorem snelpw1 4147

Description: Membership of a singleton in a unit power class. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
snelpw1	|- ({A} e. ~P1 B <-> A e. B)

Proof of Theorem snelpw1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2355	. . . 4 |- ({A} = {x} <-> {x} = {A})
2		vex 2863	. . . . 5 |- x e. _V
3	2	sneqb 3877	. . . 4 |- ({x} = {A} <-> x = A)
4	1, 3	bitri 240	. . 3 |- ({A} = {x} <-> x = A)
5	4	rexbii 2640	. 2 |- (E.x e. B {A} = {x} <-> E.x e. B x = A)
6		elpw1 4145	. 2 |- ({A} e. ~P1 B <-> E.x e. B {A} = {x})
7		risset 2662	. 2 |- (A e. B <-> E.x e. B x = A)
8	5, 6, 7	3bitr4i 268	1 |- ({A} e. ~P1 B <-> A e. B)

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  E.wrex 2616  {csn 3738  ~P₁ cpw1 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-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:  eqpw1  4163  pw1in  4165  pw10b  4167  pw1disj  4168  pw111  4171  ins2kss  4280  ins3kss  4281  ins2kexg  4306  ins3kexg  4307  dfpw2  4328  eqpw1uni  4331  ssfin  4471  ncfinraiselem2  4481  ncfinraise  4482  ncfinlower  4484  tfinsuc  4499  nnadjoinlem1  4520  sfindbl  4531  tfinnnlem1  4534  enpw1  6063  enprmaplem4  6080  nenpw1pwlem2  6086  nchoicelem11  6300  scancan  6332