![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdisjd | Structured version Visualization version GIF version |
Description: Subset preserves disjointness. Deduction form of ssdisj 4454. (Contributed by Zhi Wang, 7-Sep-2024.) |
Ref | Expression |
---|---|
ssdisjd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssdisjd.2 | ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) |
Ref | Expression |
---|---|
ssdisjd | ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdisjd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | ssrind 4230 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
3 | ssdisjd.2 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) | |
4 | sseq0 4394 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
5 | 2, 3, 4 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 |
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 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-in 3950 df-ss 3960 df-nul 4318 |
This theorem is referenced by: predisj 47766 iccdisj2 47801 sepdisj 47828 seposep 47829 |
Copyright terms: Public domain | W3C validator |