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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelrelex | Structured version Visualization version GIF version |
Description: The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5681, which could be proved from it. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelrelex | ⊢ ((Rel 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5638 | . . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | sselda 3943 | . 2 ⊢ ((Rel 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → 〈𝐴, 𝐵〉 ∈ (V × V)) |
4 | opelxp 5667 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 3, 4 | sylib 217 | 1 ⊢ ((Rel 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3909 〈cop 4591 × cxp 5629 Rel wrel 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-opab 5167 df-xp 5637 df-rel 5638 |
This theorem is referenced by: (None) |
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