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Mirrors > Home > ILE Home > Th. List > relss | GIF version |
Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
relss | ⊢ (A ⊆ B → (Rel B → Rel A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 2946 | . 2 ⊢ (A ⊆ B → (B ⊆ (V × V) → A ⊆ (V × V))) | |
2 | df-rel 4295 | . 2 ⊢ (Rel B ↔ B ⊆ (V × V)) | |
3 | df-rel 4295 | . 2 ⊢ (Rel A ↔ A ⊆ (V × V)) | |
4 | 1, 2, 3 | 3imtr4g 194 | 1 ⊢ (A ⊆ B → (Rel B → Rel A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Vcvv 2551 ⊆ wss 2911 × 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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 df-rel 4295 |
This theorem is referenced by: relin1 4398 relin2 4399 reldif 4400 relres 4582 iss 4597 cnvdif 4673 funss 4863 funssres 4885 fliftcnv 5378 fliftfun 5379 reltpos 5806 tpostpos 5820 swoer 6070 erinxp 6116 ltrel 6878 lerel 6880 |
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