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Theorem elsn 3391
 Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elsn.1 𝐴 ∈ V
Assertion
Ref Expression
elsn (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Proof of Theorem elsn
StepHypRef Expression
1 elsn.1 . 2 𝐴 ∈ V
2 elsng 3390 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2ax-mp 7 1 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381 This theorem is referenced by:  velsn  3392  sneqr  3531  onsucelsucexmid  4255  ordsoexmid  4286  opthprc  4391  dmsnm  4786  dmsnopg  4792  cnvcnvsn  4797  sniota  4894  fsn  5335  eusvobj2  5498  opelreal  6904
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