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Mirrors > Home > MPE Home > Th. List > ssdisjOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of ssdisj 3978 as of 14-Jul-2021. (Contributed by FL, 24-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ssdisjOLD | ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 3925 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) ⊆ ∅ ↔ (𝐵 ∩ 𝐶) = ∅) | |
2 | ssrin 3800 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
3 | sstr2 3575 | . . . . 5 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅)) |
5 | 1, 4 | syl5bir 232 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) = ∅ → (𝐴 ∩ 𝐶) ⊆ ∅)) |
6 | 5 | imp 444 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅) |
7 | ss0 3926 | . 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅) | |
8 | 6, 7 | syl 17 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: (None) |
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