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Theorem en1eqsnbi 8135
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 19195. (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 8134 . . 3 ((𝐴𝐵𝐵 ≈ 1𝑜) → 𝐵 = {𝐴})
21ex 450 . 2 (𝐴𝐵 → (𝐵 ≈ 1𝑜𝐵 = {𝐴}))
3 ensn1g 7965 . . 3 (𝐴𝐵 → {𝐴} ≈ 1𝑜)
4 breq1 4616 . . 3 (𝐵 = {𝐴} → (𝐵 ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜))
53, 4syl5ibrcom 237 . 2 (𝐴𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1𝑜))
62, 5impbid 202 1 (𝐴𝐵 → (𝐵 ≈ 1𝑜𝐵 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {csn 4148   class class class wbr 4613  1𝑜c1o 7498  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903
This theorem is referenced by:  srgen1zr  18451  rngen1zr  19195  rngosn4  33353
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