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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 3199 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V))) | |
| 2 | df-rel 4680 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
| 3 | df-rel 4680 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
| 4 | 1, 2, 3 | 3imtr4g 205 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 Vcvv 2771 ⊆ wss 3165 × cxp 4671 Rel wrel 4678 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-11 1528 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-in 3171 df-ss 3178 df-rel 4680 |
| This theorem is referenced by: relin1 4791 relin2 4792 reldif 4793 relres 4984 iss 5002 cnvdif 5086 funss 5287 funssres 5310 fliftcnv 5854 fliftfun 5855 reltpos 6326 tpostpos 6340 swoer 6638 erinxp 6686 ltrel 8116 lerel 8118 txdis1cn 14668 xmeter 14826 lgsquadlem1 15472 lgsquadlem2 15473 |
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