![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3979 | . 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
4 | 3 | mptru 1341 | 1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊤wtru 1333 ⊆ wss 3076 class class class wbr 3937 |
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-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 df-br 3938 |
This theorem is referenced by: brel 4599 swoer 6465 swoord1 6466 swoord2 6467 ecopover 6535 ecopoverg 6538 endom 6665 |
Copyright terms: Public domain | W3C validator |