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Theorem elsn2g 3558
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, 28-Oct-2003.)
Assertion
Ref Expression
elsn2g (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsn2g
StepHypRef Expression
1 elsni 3545 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 3554 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2202 . . 3 (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
42, 3syl5ibrcom 156 . 2 (𝐵𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
51, 4impbid2 142 1 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533
This theorem is referenced by:  elsn2  3559  elsuc2g  4327  mptiniseg  5033  elfzp1  9864  fzosplitsni  10024  zfz1isolemiso  10594
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