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Mirrors > Home > MPE Home > Th. List > ssbr | Structured version Visualization version GIF version |
Description: Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.) |
Ref | Expression |
---|---|
ssbr | ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | ssbrd 5196 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3947 class class class wbr 5153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-clel 2803 df-ss 3964 df-br 5154 |
This theorem is referenced by: ssbri 5198 coss1 5862 coss2 5863 cnvss 5879 ssrelrn 5901 ttrclss 9763 isucn2 24275 brelg 32529 cossss 38123 |
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