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Mirrors > Home > ILE Home > Th. List > brrelex12 | GIF version |
Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelex12 | ⊢ ((Rel 𝑅 ∧ A𝑅B) → (A ∈ V ∧ B ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4295 | . . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 113 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | ssbrd 3796 | . . 3 ⊢ (Rel 𝑅 → (A𝑅B → A(V × V)B)) |
4 | 3 | imp 115 | . 2 ⊢ ((Rel 𝑅 ∧ A𝑅B) → A(V × V)B) |
5 | brxp 4318 | . 2 ⊢ (A(V × V)B ↔ (A ∈ V ∧ B ∈ V)) | |
6 | 4, 5 | sylib 127 | 1 ⊢ ((Rel 𝑅 ∧ A𝑅B) → (A ∈ V ∧ B ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 class class class wbr 3755 × cxp 4286 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: brrelex 4325 brrelex2 4326 relbrcnvg 4647 ovprc 5482 ersym 6054 relelec 6082 encv 6163 |
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