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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjin2 | 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, 21-Jun-2020.) |
Ref | Expression |
---|---|
disjin2 | ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel2 4212 | . . . 4 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐶) → 𝑦 ∈ 𝐶) | |
2 | 1 | rmoimi 3751 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) |
3 | 2 | alimi 1808 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) |
4 | df-disj 5116 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
5 | df-disj 5116 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4i 292 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∈ wcel 2106 ∃*wrmo 3377 ∩ cin 3962 Disj wdisj 5115 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rmo 3378 df-v 3480 df-in 3970 df-disj 5116 |
This theorem is referenced by: ldgenpisyslem1 34144 |
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