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Theorem en1eqsnbi 9227
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20762. (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 9225 . . 3 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
21ex 414 . 2 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
3 ensn1g 8970 . . 3 (𝐴𝐵 → {𝐴} ≈ 1o)
4 breq1 5113 . . 3 (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o))
53, 4syl5ibrcom 247 . 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 1542  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:  srgen1zr  19954  rngen1zr  20762  rngosn4  36413
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