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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 4452. Alternatively this could be proved with ineqcom 4195 in tandem with ssdisjd 47704. (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 4227 | . . 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 4392 | . 2 ⊢ (((𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵) ∧ (𝐶 ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐴) = ∅) | |
6 | 3, 4, 5 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3940 ⊆ wss 3941 ∅c0 4315 |
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 2695 |
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 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 |
This theorem is referenced by: predisj 47707 seposep 47770 |
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