Theorem snel1c 4141

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

Hypothesis

Ref	Expression
snel1c.1	⊢ A ∈ V

Assertion

Ref	Expression
snel1c	⊢ {A} ∈ 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 ∈ V
3		sneq 3745	. . . . 5 ⊢ (x = A → {x} = {A})
4	3	eqeq2d 2364	. . . 4 ⊢ (x = A → ({A} = {x} ↔ {A} = {A}))
5	2, 4	spcev 2947	. . 3 ⊢ ({A} = {A} → ∃x{A} = {x})
6	1, 5	ax-mp 5	. 2 ⊢ ∃x{A} = {x}
7		el1c 4140	. 2 ⊢ ({A} ∈ 1c ↔ ∃x{A} = {x})
8	6, 7	mpbir 200	1 ⊢ {A} ∈ 1c

Syntax hints:  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {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:  snel1cg  4142  sikss1c1c  4268  ins2kss  4280  ins3kss  4281  sikexg  4297  ins2kexg  4306  ins3kexg  4307  tfin1c  4500  nnpweq  4524  sfin01  4529  pw1fnval  5852  brpw1fn  5855  df1c3  6141  tc1c  6166  ce0nn  6181  nc0le1  6217  brtcfn  6247