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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 3109 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V))) | |
2 | df-rel 4554 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
3 | df-rel 4554 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
4 | 1, 2, 3 | 3imtr4g 204 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Vcvv 2689 ⊆ wss 3076 × cxp 4545 Rel wrel 4552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 df-rel 4554 |
This theorem is referenced by: relin1 4665 relin2 4666 reldif 4667 relres 4855 iss 4873 cnvdif 4953 funss 5150 funssres 5173 fliftcnv 5704 fliftfun 5705 reltpos 6155 tpostpos 6169 swoer 6465 erinxp 6511 ltrel 7850 lerel 7852 txdis1cn 12486 xmeter 12644 |
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