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| Mirrors > Home > MPE Home > Th. List > disjss1 | Structured version Visualization version 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 3977 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 2 | 1 | anim1d 611 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
| 3 | 2 | moimdv 2546 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))) |
| 4 | 3 | alimdv 1916 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))) |
| 5 | dfdisj2 5112 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
| 6 | dfdisj2 5112 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) | |
| 7 | 4, 5, 6 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ∃*wmo 2538 ⊆ wss 3951 Disj wdisj 5110 |
| 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 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2540 df-clel 2816 df-rmo 3380 df-ss 3968 df-disj 5111 |
| This theorem is referenced by: disjeq1 5117 disjx0 5138 disjxiun 5140 disjss3 5142 volfiniun 25582 uniioovol 25614 uniioombllem4 25621 disjiunel 32609 tocyccntz 33164 carsggect 34320 carsgclctunlem2 34321 omsmeas 34325 sibfof 34342 disjf1o 45196 fsumiunss 45590 sge0iunmptlemre 46430 meadjiunlem 46480 meaiuninclem 46495 |
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