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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdisjdr | Structured version Visualization version GIF version |
Description: Subset preserves disjointness. Deduction form of ssdisj 4466. Alternatively this could be proved with ineqcom 4218 in tandem with ssdisjd 48656. (Contributed by Zhi Wang, 7-Sep-2024.) |
Ref | Expression |
---|---|
ssdisjd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssdisjdr.2 | ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅) |
Ref | Expression |
---|---|
ssdisjdr | ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdisjd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sslin 4251 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) |
4 | ssdisjdr.2 | . 2 ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅) | |
5 | sseq0 4409 | . 2 ⊢ (((𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵) ∧ (𝐶 ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐴) = ∅) | |
6 | 3, 4, 5 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅) |
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-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: predisj 48659 seposep 48722 |
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