Theorem snel1c 4140

Description: A singleton is an element of cardinal one. (Contributed by SF, 13-Jan-2015.)

Hypothesis

Ref	Expression
snel1c.1	|- A e. _V

Assertion

Ref	Expression
snel1c	|- {A} e. 1c

Proof of Theorem snel1c

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . 3 |- {A} = {A}
2		snel1c.1	. . . 4 |- A e. _V
3		sneq 3744	. . . . 5 |- (x = A -> {x} = {A})
4	3	eqeq2d 2364	. . . 4 |- (x = A -> ({A} = {x} <-> {A} = {A}))
5	2, 4	spcev 2946	. . 3 |- ({A} = {A} -> E.x{A} = {x})
6	1, 5	ax-mp 5	. 2 |- E.x{A} = {x}
7		el1c 4139	. 2 |- ({A} e. 1c <-> E.x{A} = {x})
8	6, 7	mpbir 200	1 |- {A} e. 1c

Syntax hints:  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737  1_cc1c 4134

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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136

This theorem is referenced by:  snel1cg  4141  sikss1c1c  4267  ins2kss  4279  ins3kss  4280  sikexg  4296  ins2kexg  4305  ins3kexg  4306  tfin1c  4499  nnpweq  4523  sfin01  4528  pw1fnval  5851  brpw1fn  5854  df1c3  6140  tc1c  6165  ce0nn  6180  nc0le1  6216  brtcfn  6246