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Mirrors > Home > ILE Home > Th. List > disjss1 | GIF version |
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss1 | ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . . . 6 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | anim1d 319 | . . . . 5 ⊢ (A ⊆ B → ((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶))) |
3 | 2 | alrimiv 1751 | . . . 4 ⊢ (A ⊆ B → ∀x((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶))) |
4 | moim 1961 | . . . 4 ⊢ (∀x((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶)) → (∃*x(x ∈ B ∧ y ∈ 𝐶) → ∃*x(x ∈ A ∧ y ∈ 𝐶))) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (A ⊆ B → (∃*x(x ∈ B ∧ y ∈ 𝐶) → ∃*x(x ∈ A ∧ y ∈ 𝐶))) |
6 | 5 | alimdv 1756 | . 2 ⊢ (A ⊆ B → (∀y∃*x(x ∈ B ∧ y ∈ 𝐶) → ∀y∃*x(x ∈ A ∧ y ∈ 𝐶))) |
7 | dfdisj2 3738 | . 2 ⊢ (Disj x ∈ B 𝐶 ↔ ∀y∃*x(x ∈ B ∧ y ∈ 𝐶)) | |
8 | dfdisj2 3738 | . 2 ⊢ (Disj x ∈ A 𝐶 ↔ ∀y∃*x(x ∈ A ∧ y ∈ 𝐶)) | |
9 | 6, 7, 8 | 3imtr4g 194 | 1 ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 ∈ wcel 1390 ∃*wmo 1898 ⊆ wss 2911 Disj wdisj 3736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-rmo 2308 df-in 2918 df-ss 2925 df-disj 3737 |
This theorem is referenced by: disjeq1 3743 disjx0 3754 |
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