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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 4210 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
2 | eqimss 4023 | . . 3 ⊢ ((𝐵 ∩ 𝐶) = ∅ → (𝐵 ∩ 𝐶) ⊆ ∅) | |
3 | 1, 2 | sylan9ss 3980 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅) |
4 | ss0 4352 | . 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 398 = wceq 1533 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 |
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 2156 ax-12 2172 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-nfc 2963 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 |
This theorem is referenced by: djudisj 6019 fimacnvdisj 6552 marypha1lem 8891 djuin 9341 ackbij1lem16 9651 ackbij1lem18 9653 fin23lem20 9753 fin23lem30 9758 elcls3 21685 neindisj 21719 imadifxp 30345 ldgenpisyslem1 31417 chtvalz 31895 pthhashvtx 32369 diophren 39403 |
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