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Theorem relsn 4443
 Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
Hypothesis
Ref Expression
relsn.1 𝐴 ∈ V
Assertion
Ref Expression
relsn (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsn
StepHypRef Expression
1 df-rel 4352 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 relsn.1 . . 3 𝐴 ∈ V
32snss 3494 . 2 (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
41, 3bitr4i 176 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  {csn 3375   × cxp 4343  Rel wrel 4350 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-in 2924  df-ss 2931  df-sn 3381  df-rel 4352 This theorem is referenced by:  relsnop  4444  relsn2m  4791
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