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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 3066 | . 2 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ 𝐴 → 〈𝐶, 𝐷〉 ∈ 𝐵)) |
3 | df-br 3900 | . 2 ⊢ (𝐶𝐴𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐴) | |
4 | df-br 3900 | . 2 ⊢ (𝐶𝐵𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐵) | |
5 | 2, 3, 4 | 3imtr4g 204 | 1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ⊆ wss 3041 〈cop 3500 class class class wbr 3899 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 df-br 3900 |
This theorem is referenced by: ssbri 3942 sess1 4229 brrelex12 4547 coss1 4664 coss2 4665 eqbrrdva 4679 ersym 6409 ertr 6412 |
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