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Theorem en1uniel 9061
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7748. (Revised by BTernaryTau, 24-Sep-2024.)
Assertion
Ref Expression
en1uniel (𝑆 ≈ 1o 𝑆𝑆)

Proof of Theorem en1uniel
StepHypRef Expression
1 en1b 9056 . . . 4 (𝑆 ≈ 1o𝑆 = { 𝑆})
2 eqsnuniex 5365 . . . 4 (𝑆 = { 𝑆} → 𝑆 ∈ V)
31, 2sylbi 216 . . 3 (𝑆 ≈ 1o 𝑆 ∈ V)
4 snidg 4667 . . 3 ( 𝑆 ∈ V → 𝑆 ∈ { 𝑆})
53, 4syl 17 . 2 (𝑆 ≈ 1o 𝑆 ∈ { 𝑆})
61biimpi 215 . 2 (𝑆 ≈ 1o𝑆 = { 𝑆})
75, 6eleqtrrd 2832 1 (𝑆 ≈ 1o 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  {csn 4632   cuni 4912   class class class wbr 5152  1oc1o 8488  cen 8969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1o 8495  df-en 8973
This theorem is referenced by:  en2eleq  10041  en2other2  10042  pmtrf  19424  pmtrmvd  19425  pmtrfinv  19430  frgpcyg  21521
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