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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 4359. (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 4136 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
3 | ssdisjd.2 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) | |
4 | sseq0 4298 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
5 | 2, 3, 4 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4222 |
This theorem is referenced by: predisj 45735 iccdisj2 45761 sepdisj 45787 seposep 45788 |
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