MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1eqsnbi Structured version   Visualization version   GIF version

Theorem en1eqsnbi 9308
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20795. (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 9306 . . 3 ((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
21ex 412 . 2 (𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
3 ensn1g 9061 . . 3 (𝐴𝐵 → {𝐴} ≈ 1o)
4 breq1 5151 . . 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 1537  wcel 2106  {csn 4631   class class class wbr 5148  1oc1o 8498  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1o 8505  df-en 8985
This theorem is referenced by:  srgen1zr  20234  rngen1zr  20795  rngosn4  37912
  Copyright terms: Public domain W3C validator