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Theorem snelsingles 32004
Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
Hypothesis
Ref Expression
snelsingles.1 𝐴 ∈ V
Assertion
Ref Expression
snelsingles {𝐴} ∈ Singletons

Proof of Theorem snelsingles
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 snelsingles.1 . . . 4 𝐴 ∈ V
2 isset 3202 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 eqcom 2627 . . . . . 6 (𝑥 = 𝐴𝐴 = 𝑥)
43exbii 1772 . . . . 5 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
52, 4bitri 264 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
61, 5mpbi 220 . . 3 𝑥 𝐴 = 𝑥
7 sneq 4178 . . 3 (𝐴 = 𝑥 → {𝐴} = {𝑥})
86, 7eximii 1762 . 2 𝑥{𝐴} = {𝑥}
9 elsingles 32000 . 2 ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
108, 9mpbir 221 1 {𝐴} ∈ Singletons
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  {csn 4168   Singletons csingles 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-symdif 3836  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-eprel 5019  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-1st 7153  df-2nd 7154  df-txp 31935  df-singleton 31943  df-singles 31944
This theorem is referenced by: (None)
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