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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 20756. (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 411 . 2 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
3 ensn1g 9055 . . 3 (𝐴𝐵 → {𝐴} ≈ 1o)
4 breq1 5156 . . 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 1534  wcel 2099  {csn 4633   class class class wbr 5153  1oc1o 8489  cen 8971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-1o 8496  df-en 8975
This theorem is referenced by:  srgen1zr  20199  rngen1zr  20756  rngosn4  37626
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