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Theorem bj-opelrelex 34452
 Description: The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5577, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
Assertion
Ref Expression
bj-opelrelex ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem bj-opelrelex
StepHypRef Expression
1 df-rel 5535 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 219 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3943 . 2 ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4 opelxp 5564 . 2 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
53, 4sylib 221 1 ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  Vcvv 3471   ⊆ wss 3910  ⟨cop 4546   × cxp 5526  Rel wrel 5533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-opab 5102  df-xp 5534  df-rel 5535 This theorem is referenced by: (None)
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