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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 5597 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 class class class wbr 5059 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 |
This theorem is referenced by: nprrel12 5603 vtoclr 5608 relbrcnvg 5961 ovprc 7187 oprabv 7207 encv 8510 fsuppimp 8832 fsuppunbi 8847 brfi1uzind 13853 brfi1indALT 13855 isstruct2 16486 brssc 17077 isfull 17173 isfth 17177 dvdsr 19389 ulmval 24964 subgrv 27048 vcex 28351 opelco3 33037 bj-ideqgALT 34472 bj-idreseqb 34477 bj-ideqg1ALT 34479 rngoablo2 35221 aovprc 43462 aovrcl 43463 nelbrim 43549 isisomgr 44059 linindsv 44570 |
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