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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdisjd | Structured version Visualization version GIF version |
Description: Subset preserves disjointness. Deduction form of ssdisj 4466. (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 4252 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
3 | ssdisjd.2 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) | |
4 | sseq0 4409 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
5 | 2, 3, 4 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: predisj 48659 iccdisj2 48694 sepdisj 48721 seposep 48722 |
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