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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 3091 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 334 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
3 | 2 | alrimiv 1846 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
4 | moim 2063 | . . . 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 1851 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))) |
7 | dfdisj2 3908 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
8 | dfdisj2 3908 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) | |
9 | 6, 7, 8 | 3imtr4g 204 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 ∈ wcel 1480 ∃*wmo 2000 ⊆ wss 3071 Disj wdisj 3906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-rmo 2424 df-in 3077 df-ss 3084 df-disj 3907 |
This theorem is referenced by: disjeq1 3913 disjx0 3928 fsumiun 11246 |
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