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Theorem elsn2 4187
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
elsn2.1 𝐵 ∈ V
Assertion
Ref Expression
elsn2 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Proof of Theorem elsn2
StepHypRef Expression
1 elsn2.1 . 2 𝐵 ∈ V
2 elsn2g 4186 . 2 (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2ax-mp 5 1 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-sn 4154
This theorem is referenced by:  fparlem1  7223  fparlem2  7224  el1o  7525  fin1a2lem11  9177  fin1a2lem12  9178  elnn0  11239  elxnn0  11310  elfzp1  12330  fsumss  14384  fprodss  14598  elhoma  16598  islpidl  19160  zrhrhmb  19773  rest0  20878  qustgphaus  21831  taylfval  24012  elch0  27951  atoml2i  29082  bj-eltag  32604  bj-rest10b  32671  dibopelvalN  35898  dibopelval2  35900  climrec  39226
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