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Theorem en1uniel 7914
 Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
en1uniel (𝑆 ≈ 1𝑜 𝑆𝑆)

Proof of Theorem en1uniel
StepHypRef Expression
1 relen 7846 . . . 4 Rel ≈
21brrelexi 5082 . . 3 (𝑆 ≈ 1𝑜𝑆 ∈ V)
3 uniexg 6853 . . 3 (𝑆 ∈ V → 𝑆 ∈ V)
4 snidg 4153 . . 3 ( 𝑆 ∈ V → 𝑆 ∈ { 𝑆})
52, 3, 43syl 18 . 2 (𝑆 ≈ 1𝑜 𝑆 ∈ { 𝑆})
6 en1b 7910 . . 3 (𝑆 ≈ 1𝑜𝑆 = { 𝑆})
76biimpi 205 . 2 (𝑆 ≈ 1𝑜𝑆 = { 𝑆})
85, 7eleqtrrd 2691 1 (𝑆 ≈ 1𝑜 𝑆𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ∪ cuni 4372   class class class wbr 4583  1𝑜c1o 7440   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-en 7842 This theorem is referenced by:  en2eleq  8714  en2other2  8715  pmtrf  17698  pmtrmvd  17699  pmtrfinv  17704  frgpcyg  19741
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