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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 2938 | . 2 ⊢ (φ → (〈𝐶, 𝐷〉 ∈ A → 〈𝐶, 𝐷〉 ∈ B)) |
3 | df-br 3756 | . 2 ⊢ (𝐶A𝐷 ↔ 〈𝐶, 𝐷〉 ∈ A) | |
4 | df-br 3756 | . 2 ⊢ (𝐶B𝐷 ↔ 〈𝐶, 𝐷〉 ∈ B) | |
5 | 2, 3, 4 | 3imtr4g 194 | 1 ⊢ (φ → (𝐶A𝐷 → 𝐶B𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 df-br 3756 |
This theorem is referenced by: ssbri 3797 sess1 4059 brrelex12 4324 coss1 4434 coss2 4435 eqbrrdva 4448 ersym 6054 ertr 6057 |
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