Theorem snelpw1 4146

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 2862	. . . . 5 |- x e. _V
3	2	sneqb 3876	. . . 4 |- ({x} = {A} <-> x = A)
4	1, 3	bitri 240	. . 3 |- ({A} = {x} <-> x = A)
5	4	rexbii 2639	. 2 |- (E.x e. B {A} = {x} <-> E.x e. B x = A)
6		elpw1 4144	. 2 |- ({A} e. ~P1 B <-> E.x e. B {A} = {x})
7		risset 2661	. 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 2615  {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-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:  eqpw1  4162  pw1in  4164  pw10b  4166  pw1disj  4167  pw111  4170  ins2kss  4279  ins3kss  4280  ins2kexg  4305  ins3kexg  4306  dfpw2  4327  eqpw1uni  4330  ssfin  4470  ncfinraiselem2  4480  ncfinraise  4481  ncfinlower  4483  tfinsuc  4498  nnadjoinlem1  4519  sfindbl  4530  tfinnnlem1  4533  enpw1  6062  enprmaplem4  6079  nenpw1pwlem2  6085  nchoicelem11  6299  scancan  6331