MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1uniel Structured version   Visualization version   GIF version

Theorem en1uniel 9030
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7722. (Revised by BTernaryTau, 24-Sep-2024.)
Assertion
Ref Expression
en1uniel (𝑆 ≈ 1o 𝑆𝑆)

Proof of Theorem en1uniel
StepHypRef Expression
1 en1b 9025 . . . 4 (𝑆 ≈ 1o𝑆 = { 𝑆})
2 eqsnuniex 5352 . . . 4 (𝑆 = { 𝑆} → 𝑆 ∈ V)
31, 2sylbi 216 . . 3 (𝑆 ≈ 1o 𝑆 ∈ V)
4 snidg 4657 . . 3 ( 𝑆 ∈ V → 𝑆 ∈ { 𝑆})
53, 4syl 17 . 2 (𝑆 ≈ 1o 𝑆 ∈ { 𝑆})
61biimpi 215 . 2 (𝑆 ≈ 1o𝑆 = { 𝑆})
75, 6eleqtrrd 2830 1 (𝑆 ≈ 1o 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623   cuni 4902   class class class wbr 5141  1oc1o 8460  cen 8938
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8467  df-en 8942
This theorem is referenced by:  en2eleq  10005  en2other2  10006  pmtrf  19375  pmtrmvd  19376  pmtrfinv  19381  frgpcyg  21468
  Copyright terms: Public domain W3C validator