Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 | ssrmo 28718 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
4 | 3 | alimdv 1832 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
5 | df-disj 4554 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
6 | df-disj 4554 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
7 | 4, 5, 6 | 3imtr4g 284 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∈ wcel 1977 Ⅎwnfc 2738 ∃*wrmo 2899 ⊆ wss 3540 Disj wdisj 4553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rmo 2904 df-in 3547 df-ss 3554 df-disj 4554 |
This theorem is referenced by: disjeq1f 28769 esumrnmpt2 29457 measvuni 29604 |
Copyright terms: Public domain | W3C validator |