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Mirrors > Home > MPE Home > Th. List > disjss2 | Structured version Visualization version GIF version |
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss2 | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3961 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
2 | 1 | ralimi 3160 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
3 | rmoim 3731 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
5 | 4 | alimdv 1917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
6 | df-disj 5032 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
7 | df-disj 5032 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3imtr4g 298 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∈ wcel 2114 ∀wral 3138 ∃*wrmo 3141 ⊆ wss 3936 Disj wdisj 5031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rmo 3146 df-in 3943 df-ss 3952 df-disj 5032 |
This theorem is referenced by: disjeq2 5035 0disj 5058 uniioombllem2 24184 uniioombllem4 24187 disjxwwlksn 27682 disjxwwlkn 27692 fusgreghash2wspv 28114 fsumiunss 41876 |
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