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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 3799 | . . . . . 6 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶) | |
2 | 1 | anim2i 593 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐶 ∩ 𝐴)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
3 | 2 | ax-gen 1722 | . . . 4 ⊢ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐶 ∩ 𝐴)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
4 | 3 | rmoimi2 3409 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
5 | 4 | alimi 1739 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
6 | df-disj 4621 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
7 | df-disj 4621 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) | |
8 | 5, 6, 7 | 3imtr4i 281 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∃*wrmo 2915 ∩ cin 3573 Disj wdisj 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rmo 2920 df-v 3202 df-in 3581 df-disj 4621 |
This theorem is referenced by: measinblem 30283 carsgclctunlem2 30381 |
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