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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 3177 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V))) | |
2 | df-rel 4648 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
3 | df-rel 4648 | . 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 2752 ⊆ wss 3144 × cxp 4639 Rel wrel 4646 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 df-rel 4648 |
This theorem is referenced by: relin1 4759 relin2 4760 reldif 4761 relres 4950 iss 4968 cnvdif 5050 funss 5251 funssres 5274 fliftcnv 5813 fliftfun 5814 reltpos 6270 tpostpos 6284 swoer 6582 erinxp 6630 ltrel 8044 lerel 8046 txdis1cn 14215 xmeter 14373 |
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