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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 5736 | . 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 2107 Vcvv 3479 class class class wbr 5142 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: nprrel12 5742 vtoclr 5747 relbrcnvg 6122 ovprc 7470 oprabv 7494 encv 8994 brdomi 9000 domssl 9039 fsuppimp 9409 fsuppunbi 9430 brttrcl 9754 brfi1uzind 14548 brfi1indALT 14550 isstruct2 17187 brssc 17859 isfull 17958 isfth 17962 dvdsr 20363 ulmval 26424 subgrv 29288 vcex 30598 opelco3 35776 bj-ideqgALT 37160 bj-idreseqb 37165 bj-ideqg1ALT 37167 rngoablo2 37917 aovprc 47205 aovrcl 47206 nelbrim 47292 linindsv 48367 upfval3 48953 | 
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