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Mirrors > Home > MPE Home > Th. List > ssbrd | Structured version Visualization version 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 3567 | . 2 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ 𝐴 → 〈𝐶, 𝐷〉 ∈ 𝐵)) |
3 | df-br 4584 | . 2 ⊢ (𝐶𝐴𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐴) | |
4 | df-br 4584 | . 2 ⊢ (𝐶𝐵𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐵) | |
5 | 2, 3, 4 | 3imtr4g 284 | 1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-br 4584 |
This theorem is referenced by: ssbri 4627 sess1 5006 brrelex12 5079 coss1 5199 coss2 5200 eqbrrdva 5213 cnvss 5216 ssrelrn 5237 ersym 7641 ertr 7644 fpwwe2lem6 9336 fpwwe2lem7 9337 fpwwe2lem9 9339 fpwwe2lem12 9342 fpwwe2lem13 9343 fpwwe2 9344 coss12d 13559 fthres2 16415 invfuc 16457 pospo 16796 dirref 17058 efgcpbl 17992 frgpuplem 18008 subrguss 18618 znleval 19722 ustref 21832 ustuqtop4 21858 isucn2 21893 brelg 28801 metider 29265 mclsppslem 30734 fundmpss 30910 iunrelexpuztr 37030 frege96d 37060 frege91d 37062 frege98d 37064 frege124d 37072 |
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