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Theorem en1eqsn 9225
Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5325, ax-un 7677. (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 8972 . . 3 (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2 eleq2 2827 . . . . . . . 8 (𝐵 = {𝑥} → (𝐴𝐵𝐴 ∈ {𝑥}))
3 elsni 4608 . . . . . . . . 9 (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
43sneqd 4603 . . . . . . . 8 (𝐴 ∈ {𝑥} → {𝐴} = {𝑥})
52, 4syl6bi 253 . . . . . . 7 (𝐵 = {𝑥} → (𝐴𝐵 → {𝐴} = {𝑥}))
65imp 408 . . . . . 6 ((𝐵 = {𝑥} ∧ 𝐴𝐵) → {𝐴} = {𝑥})
7 eqtr3 2763 . . . . . 6 ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
86, 7syldan 592 . . . . 5 ((𝐵 = {𝑥} ∧ 𝐴𝐵) → 𝐵 = {𝐴})
98ex 414 . . . 4 (𝐵 = {𝑥} → (𝐴𝐵𝐵 = {𝐴}))
109exlimiv 1934 . . 3 (∃𝑥 𝐵 = {𝑥} → (𝐴𝐵𝐵 = {𝐴}))
111, 10sylbi 216 . 2 (𝐵 ≈ 1o → (𝐴𝐵𝐵 = {𝐴}))
1211impcom 409 1 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  {csn 4591   class class class wbr 5110  1oc1o 8410  cen 8887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1o 8417  df-en 8891
This theorem is referenced by:  en1eqsnbi  9227  1nsgtrivd  18983  gex1  19380  0cyg  19677  pgpfac1lem3a  19862  pgpfaclem3  19869  0ring  20756  en1top  22350  cnextfres1  23435  xrge0tsmseq  31943  sconnpi1  33873  rngoueqz  36428  isdmn3  36562
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