Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snelsingles Structured version   Visualization version   GIF version

Theorem snelsingles 32542
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 3395 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 eqcom 2806 . . . . . 6 (𝑥 = 𝐴𝐴 = 𝑥)
43exbii 1944 . . . . 5 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
52, 4bitri 267 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
61, 5mpbi 222 . . 3 𝑥 𝐴 = 𝑥
7 sneq 4378 . . 3 (𝐴 = 𝑥 → {𝐴} = {𝑥})
86, 7eximii 1932 . 2 𝑥{𝐴} = {𝑥}
9 elsingles 32538 . 2 ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
108, 9mpbir 223 1 {𝐴} ∈ Singletons
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  {csn 4368   Singletons csingles 32459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-symdif 4041  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-eprel 5225  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-txp 32474  df-singleton 32482  df-singles 32483
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator