Theorem dfsingles2 36122

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 36119	. 2 ⊢ (𝑥 ∈ Singletons ↔ ∃𝑦 𝑥 = {𝑦})
2	1	eqabi 2872	1 ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}

Syntax hints:   = wceq 1542  ∃wex 1781  {cab 2715  {csn 4568   Singletons csingles 36040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7933  df-2nd 7934  df-txp 36055  df-singleton 36063  df-singles 36064

This theorem is referenced by:  dfiota3  36124