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Theorem elsn2 3436
 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 3435 . 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 1285   ∈ wcel 1434  Vcvv 2602  {csn 3406 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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sn 3412 This theorem is referenced by:  el1o  6084  elnn0  8357  elxnn0  8420
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