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| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjss1f | Structured version Visualization version GIF version | ||
| Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
| Ref | Expression |
|---|---|
| disjss1f.1 | ⊢ Ⅎ𝑥𝐴 |
| disjss1f.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| disjss1f | ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | disjss1f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | disjss1f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | ssrmof 4050 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
| 4 | 3 | alimdv 1916 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
| 5 | df-disj 5109 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
| 6 | df-disj 5109 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*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 ∀wal 1538 ∈ wcel 2108 Ⅎwnfc 2889 ∃*wrmo 3378 ⊆ wss 3950 Disj wdisj 5108 |
| 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 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2539 df-clel 2815 df-nfc 2891 df-rmo 3379 df-ss 3967 df-disj 5109 |
| This theorem is referenced by: disjeq1f 32575 esumrnmpt2 34047 measvuni 34193 |
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