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Theorem en1eqsnbi 9310
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20778. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
Assertion
Ref Expression
en1eqsnbi (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))

Proof of Theorem en1eqsnbi
StepHypRef Expression
1 en1eqsn 9308 . . 3 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
21ex 412 . 2 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
3 ensn1g 9062 . . 3 (𝐴𝐵 → {𝐴} ≈ 1o)
4 breq1 5146 . . 3 (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o))
53, 4syl5ibrcom 247 . 2 (𝐴𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1o))
62, 5impbid 212 1 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {csn 4626   class class class wbr 5143  1oc1o 8499  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986
This theorem is referenced by:  srgen1zr  20213  rngen1zr  20778  rngosn4  37932
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