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Mirrors > Home > ILE Home > Th. List > ssbrd | GIF version |
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
ssbrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssbrd | ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | sseld 2998 | . 2 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ 𝐴 → 〈𝐶, 𝐷〉 ∈ 𝐵)) |
3 | df-br 3786 | . 2 ⊢ (𝐶𝐴𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐴) | |
4 | df-br 3786 | . 2 ⊢ (𝐶𝐵𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐵) | |
5 | 2, 3, 4 | 3imtr4g 203 | 1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 ⊆ wss 2973 〈cop 3401 class class class wbr 3785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-in 2979 df-ss 2986 df-br 3786 |
This theorem is referenced by: ssbri 3827 sess1 4092 brrelex12 4399 coss1 4509 coss2 4510 eqbrrdva 4523 ersym 6141 ertr 6144 |
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