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Theorem snelsingles 35903
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 3491 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 eqcom 2741 . . . . . 6 (𝑥 = 𝐴𝐴 = 𝑥)
43exbii 1844 . . . . 5 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
52, 4bitri 275 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
61, 5mpbi 230 . . 3 𝑥 𝐴 = 𝑥
7 sneq 4640 . . 3 (𝐴 = 𝑥 → {𝐴} = {𝑥})
86, 7eximii 1833 . 2 𝑥{𝐴} = {𝑥}
9 elsingles 35899 . 2 ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
108, 9mpbir 231 1 {𝐴} ∈ Singletons
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  {csn 4630   Singletons csingles 35820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-symdif 4258  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-singleton 35843  df-singles 35844
This theorem is referenced by: (None)
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