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Theorem elsng 3607
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
elsng (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2184 . 2 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
2 df-sn 3598 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
31, 2elab2g 2884 1 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sn 3598
This theorem is referenced by:  elsn  3608  elsni  3610  snidg  3621  eltpg  3637  eldifsn  3719  elsucg  4403  funconstss  5633  fniniseg  5635  fniniseg2  5637  fidcenumlemrks  6949  ltxr  9771  elfzp12  10094  1exp  10544  0subm  12803  0subg  12990  0nsg  13005
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