Theorem dfsingles2 33386

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

Step	Hyp	Ref	Expression
1		elsingles 33383	. 2 ⊢ (𝑥 ∈ Singletons ↔ ∃𝑦 𝑥 = {𝑦})
2	1	abbi2i 2956	1 ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}

Syntax hints:   = wceq 1536  ∃wex 1779  {cab 2802  {csn 4570   Singletons csingles 33304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-eprel 5468  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-1st 7692  df-2nd 7693  df-txp 33319  df-singleton 33327  df-singles 33328

This theorem is referenced by:  dfiota3  33388