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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 4234 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
2 | eqimss 4041 | . . 3 ⊢ ((𝐵 ∩ 𝐶) = ∅ → (𝐵 ∩ 𝐶) ⊆ ∅) | |
3 | 1, 2 | sylan9ss 3996 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅) |
4 | ss0 4399 | . 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 397 = wceq 1542 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 |
This theorem is referenced by: djudisj 6167 fimacnvdisj 6770 marypha1lem 9428 djuin 9913 ackbij1lem16 10230 ackbij1lem18 10232 fin23lem20 10332 fin23lem30 10337 elcls3 22587 neindisj 22621 imadifxp 31832 ldgenpisyslem1 33161 chtvalz 33641 pthhashvtx 34118 diophren 41551 |
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