Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfsingles2 Structured version   Visualization version   GIF version

Theorem dfsingles2 32541
 Description: Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfsingles2 Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfsingles2
StepHypRef Expression
1 elsingles 32538 . 2 (𝑥 Singletons ↔ ∃𝑦 𝑥 = {𝑦})
21abbi2i 2915 1 Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1653  ∃wex 1875  {cab 2785  {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:  dfiota3  32543
 Copyright terms: Public domain W3C validator