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Theorem snelsingles 35441
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 3482 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 eqcom 2734 . . . . . 6 (𝑥 = 𝐴𝐴 = 𝑥)
43exbii 1843 . . . . 5 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
52, 4bitri 275 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
61, 5mpbi 229 . . 3 𝑥 𝐴 = 𝑥
7 sneq 4634 . . 3 (𝐴 = 𝑥 → {𝐴} = {𝑥})
86, 7eximii 1832 . 2 𝑥{𝐴} = {𝑥}
9 elsingles 35437 . 2 ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
108, 9mpbir 230 1 {𝐴} ∈ Singletons
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  wcel 2099  Vcvv 3469  {csn 4624   Singletons csingles 35358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-symdif 4238  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7985  df-2nd 7986  df-txp 35373  df-singleton 35381  df-singles 35382
This theorem is referenced by: (None)
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