Theorem snel1cg 4142

Description: A singleton is an element of cardinal one. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
snel1cg	|- (A e. V -> {A} e. 1c)

Proof of Theorem snel1cg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . 3 |- (x = A -> {x} = {A})
2	1	eleq1d 2419	. 2 |- (x = A -> ({x} e. 1c <-> {A} e. 1c))
3		vex 2863	. . 3 |- x e. _V
4	3	snel1c 4141	. 2 |- {x} e. 1c
5	2, 4	vtoclg 2915	1 |- (A e. V -> {A} e. 1c)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {csn 3738  1_cc1c 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 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137

This theorem is referenced by:  snfi  4432  ncfinsn  4477  nchoicelem13  6302