ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relsn GIF version

Theorem relsn 4470
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 4379 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 relsn.1 . . 3 𝐴 ∈ V
32snss 3521 . 2 (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
41, 3bitr4i 180 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  Vcvv 2574  wss 2944  {csn 3402   × cxp 4370  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-sn 3408  df-rel 4379
This theorem is referenced by:  relsnop  4471  relsn2m  4818
  Copyright terms: Public domain W3C validator