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Theorem elrel 4681
 Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
elrel ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem elrel
StepHypRef Expression
1 df-rel 4586 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 119 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3124 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 elvv 4641 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
53, 4sylib 121 1 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2125  Vcvv 2709   ⊆ wss 3098  ⟨cop 3559   × cxp 4577  Rel wrel 4584 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-opab 4022  df-xp 4585  df-rel 4586 This theorem is referenced by:  eliunxp  4718  elres  4895  unielrel  5106  rntpos  6194
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