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| Mirrors > Home > MPE Home > Th. List > ssbri | Structured version Visualization version 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 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
| 2 | ssbr 5187 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 df-ss 3968 df-br 5144 |
| This theorem is referenced by: brel 5750 swoer 8776 swoord1 8777 swoord2 8778 ecopover 8861 endom 9019 brdom3 10568 brdom5 10569 brdom4 10570 fpwwe2lem12 10682 nqerf 10970 nqerrel 10972 isfull 17957 isfth 17961 fulloppc 17969 fthoppc 17970 fthsect 17972 fthinv 17973 fthmon 17974 fthepi 17975 ffthiso 17976 catcisolem 18155 psss 18625 efgrelex 19769 hlimadd 31212 hhsscms 31297 occllem 31322 nlelchi 32080 hmopidmchi 32170 fundmpss 35767 itg2gt0cn 37682 brresi2 37727 fullthinc2 49100 thincciso 49102 |
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