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Theorem en1eqsnbi 9275
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20397. (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 9273 . . 3 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
21ex 413 . 2 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
3 ensn1g 9018 . . 3 (𝐴𝐵 → {𝐴} ≈ 1o)
4 breq1 5151 . . 3 (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o))
53, 4syl5ibrcom 246 . 2 (𝐴𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1o))
62, 5impbid 211 1 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  {csn 4628   class class class wbr 5148  1oc1o 8458  cen 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8465  df-en 8939
This theorem is referenced by:  srgen1zr  20038  rngen1zr  20397  rngosn4  36788
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