Theorem dfsingles2 35970

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 35967	. 2 ⊢ (𝑥 ∈ Singletons ↔ ∃𝑦 𝑥 = {𝑦})
2	1	eqabi 2866	1 ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}

Syntax hints:   = wceq 1541  ∃wex 1780  {cab 2709  {csn 4575   Singletons csingles 35888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-symdif 4202  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-1st 7927  df-2nd 7928  df-txp 35903  df-singleton 35911  df-singles 35912

This theorem is referenced by:  dfiota3  35972