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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjin | Structured version Visualization version GIF version |
Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
Ref | Expression |
---|---|
disjin | ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel1 4190 | . . . 4 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶) | |
2 | 1 | rmoimi 3733 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
3 | 2 | alimi 1805 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
4 | df-disj 5107 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
5 | df-disj 5107 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 292 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2098 ∃*wrmo 3369 ∩ cin 3942 Disj wdisj 5106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rmo 3370 df-v 3470 df-in 3950 df-disj 5107 |
This theorem is referenced by: measinblem 33748 carsgclctunlem2 33848 |
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