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Mirrors > Home > MPE Home > Th. List > ssdisj | Structured version Visualization version GIF version |
Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
ssdisj | ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4034 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
2 | eqimss 3854 | . . 3 ⊢ ((𝐵 ∩ 𝐶) = ∅ → (𝐵 ∩ 𝐶) ⊆ ∅) | |
3 | 1, 2 | sylan9ss 3812 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅) |
4 | ss0 4171 | . 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∩ cin 3769 ⊆ wss 3770 ∅c0 4116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-v 3388 df-dif 3773 df-in 3777 df-ss 3784 df-nul 4117 |
This theorem is referenced by: djudisj 5779 fimacnvdisj 6299 marypha1lem 8582 djuin 9031 ackbij1lem16 9346 ackbij1lem18 9348 fin23lem20 9448 fin23lem30 9453 elcls3 21215 neindisj 21249 imadifxp 29930 ldgenpisyslem1 30741 chtvalz 31226 diophren 38158 |
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