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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 4021 | . . . . . 6 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶) | |
2 | 1 | anim2i 610 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐶 ∩ 𝐴)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
3 | 2 | ax-gen 1839 | . . . 4 ⊢ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐶 ∩ 𝐴)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
4 | 3 | rmoimi2 3622 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
5 | 4 | alimi 1855 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
6 | df-disj 4855 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
7 | df-disj 4855 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) | |
8 | 5, 6, 7 | 3imtr4i 284 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1599 ∈ wcel 2106 ∃*wrmo 3092 ∩ cin 3790 Disj wdisj 4854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rmo 3097 df-v 3399 df-in 3798 df-disj 4855 |
This theorem is referenced by: measinblem 30881 carsgclctunlem2 30979 |
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