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Mirrors > Home > ILE Home > Th. List > ssbri | GIF version |
Description: Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
ssbri.1 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
ssbri | ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbri.1 | . . . 4 ⊢ 𝐴 ⊆ 𝐵 | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → 𝐴 ⊆ 𝐵) |
3 | 2 | ssbrd 3971 | . 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
4 | 3 | mptru 1340 | 1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊤wtru 1332 ⊆ wss 3071 class class class wbr 3929 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 df-br 3930 |
This theorem is referenced by: brel 4591 swoer 6457 swoord1 6458 swoord2 6459 ecopover 6527 ecopoverg 6530 endom 6657 |
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