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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 4096 | . . . 4 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐶) → 𝑦 ∈ 𝐶) | |
2 | 1 | rmoimi 3644 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) |
3 | 2 | alimi 1819 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) |
4 | df-disj 5005 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
5 | df-disj 5005 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4i 295 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∈ wcel 2112 ∃*wrmo 3054 ∩ cin 3852 Disj wdisj 5004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-mo 2539 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rmo 3059 df-v 3400 df-in 3860 df-disj 5005 |
This theorem is referenced by: ldgenpisyslem1 31797 |
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