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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 5101 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3935 class class class wbr 5058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 df-br 5059 |
This theorem is referenced by: ssbri 5103 coss1 5720 coss2 5721 cnvss 5737 ssrelrn 5757 isucn2 22882 brelg 30354 cossss 35664 |
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