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Mirrors > Home > MPE Home > Th. List > brrelex12i | Structured version Visualization version GIF version |
Description: Two classes that are related by a binary relation are sets. Inference form. (Contributed by BJ, 3-Oct-2022.) |
Ref | Expression |
---|---|
brrelexi.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
brrelex12i | ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelexi.1 | . 2 ⊢ Rel 𝑅 | |
2 | brrelex12 5741 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: nprrel12 5747 vtoclr 5752 relbrcnvg 6126 ovprc 7469 oprabv 7493 encv 8992 brdomi 8998 domssl 9037 fsuppimp 9406 fsuppunbi 9427 brttrcl 9751 brfi1uzind 14544 brfi1indALT 14546 isstruct2 17183 brssc 17862 isfull 17964 isfth 17968 dvdsr 20379 ulmval 26438 subgrv 29302 vcex 30607 opelco3 35756 bj-ideqgALT 37141 bj-idreseqb 37146 bj-ideqg1ALT 37148 rngoablo2 37896 aovprc 47138 aovrcl 47139 nelbrim 47225 linindsv 48291 |
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