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Theorem elsn2 3463
Description: There is only 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 3462 . 2 (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2ax-mp 7 1 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1287  wcel 1436  Vcvv 2615  {csn 3431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sn 3437
This theorem is referenced by:  el1o  6157  elnn0  8611  elxnn0  8674  fisumss  10675
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