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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 3145 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V))) | |
2 | df-rel 4606 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
3 | df-rel 4606 | . 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 2722 ⊆ wss 3112 × cxp 4597 Rel wrel 4604 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3118 df-ss 3125 df-rel 4606 |
This theorem is referenced by: relin1 4717 relin2 4718 reldif 4719 relres 4907 iss 4925 cnvdif 5005 funss 5202 funssres 5225 fliftcnv 5758 fliftfun 5759 reltpos 6210 tpostpos 6224 swoer 6521 erinxp 6567 ltrel 7952 lerel 7954 txdis1cn 12845 xmeter 13003 |
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