Theorem bj-opelrelex 37816

Description: The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5712, 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

Step	Hyp	Ref	Expression
1		df-rel 5667	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 219	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3936	. 2 ⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4		opelxp 5696	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	3, 4	sylib 221	1 ⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  ⟨cop 4594   × cxp 5658  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-xp 5666  df-rel 5667

This theorem is used by: (None)