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Theorem en1eqsn 9306
Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5371, ax-un 7754. (Revised by BTernaryTau, 4-Jan-2025.)
Assertion
Ref Expression
en1eqsn ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})

Proof of Theorem en1eqsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 en1 9063 . . 3 (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2 eleq2 2828 . . . . . . . 8 (𝐵 = {𝑥} → (𝐴𝐵𝐴 ∈ {𝑥}))
3 elsni 4648 . . . . . . . . 9 (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
43sneqd 4643 . . . . . . . 8 (𝐴 ∈ {𝑥} → {𝐴} = {𝑥})
52, 4biimtrdi 253 . . . . . . 7 (𝐵 = {𝑥} → (𝐴𝐵 → {𝐴} = {𝑥}))
65imp 406 . . . . . 6 ((𝐵 = {𝑥} ∧ 𝐴𝐵) → {𝐴} = {𝑥})
7 eqtr3 2761 . . . . . 6 ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
86, 7syldan 591 . . . . 5 ((𝐵 = {𝑥} ∧ 𝐴𝐵) → 𝐵 = {𝐴})
98ex 412 . . . 4 (𝐵 = {𝑥} → (𝐴𝐵𝐵 = {𝐴}))
109exlimiv 1928 . . 3 (∃𝑥 𝐵 = {𝑥} → (𝐴𝐵𝐵 = {𝐴}))
111, 10sylbi 217 . 2 (𝐵 ≈ 1o → (𝐴𝐵𝐵 = {𝐴}))
1211impcom 407 1 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {csn 4631   class class class wbr 5148  1oc1o 8498  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1o 8505  df-en 8985
This theorem is referenced by:  en1eqsnbi  9308  1nsgtrivd  19205  gex1  19624  0cyg  19926  pgpfac1lem3a  20111  pgpfaclem3  20118  0ring  20543  en1top  23007  cnextfres1  24092  xrge0tsmseq  33050  sconnpi1  35224  rngoueqz  37927  isdmn3  38061
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