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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 3970 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 609 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
3 | 2 | moimdv 2534 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))) |
4 | 3 | alimdv 1911 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))) |
5 | dfdisj2 5116 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
6 | dfdisj2 5116 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) | |
7 | 4, 5, 6 | 3imtr4g 295 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 ∈ wcel 2098 ∃*wmo 2526 ⊆ wss 3944 Disj wdisj 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-mo 2528 df-clel 2802 df-rmo 3363 df-ss 3961 df-disj 5115 |
This theorem is referenced by: disjeq1 5121 disjx0 5143 disjxiun 5146 disjss3 5148 volfiniun 25520 uniioovol 25552 uniioombllem4 25559 disjiunel 32465 tocyccntz 32957 carsggect 34069 carsgclctunlem2 34070 omsmeas 34074 sibfof 34091 disjf1o 44703 fsumiunss 45101 sge0iunmptlemre 45941 meadjiunlem 45991 meaiuninclem 46006 |
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