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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 4175 | . . . 4 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶) | |
2 | 1 | rmoimi 3736 | . . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
3 | 2 | alimi 1811 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) |
4 | df-disj 5035 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
5 | df-disj 5035 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 294 | 1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∈ wcel 2113 ∃*wrmo 3144 ∩ cin 3938 Disj wdisj 5034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rmo 3149 df-v 3499 df-in 3946 df-disj 5035 |
This theorem is referenced by: measinblem 31483 carsgclctunlem2 31581 |
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