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Mirrors > Home > HSE Home > Th. List > cvnbtwn4 | Structured version Visualization version GIF version |
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvnbtwn4 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvnbtwn 29701 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 392 | . . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
3 | an4 648 | . . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) | |
4 | ioran 1013 | . . . . . . 7 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵)) | |
5 | eqcom 2833 | . . . . . . . . 9 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
6 | 5 | notbii 312 | . . . . . . . 8 ⊢ (¬ 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶) |
7 | 6 | anbi1i 619 | . . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) |
8 | 4, 7 | bitri 267 | . . . . . 6 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) |
9 | 8 | anbi2i 618 | . . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))) |
10 | dfpss2 3919 | . . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
11 | dfpss2 3919 | . . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)) | |
12 | 10, 11 | anbi12i 622 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) |
13 | 3, 9, 12 | 3bitr4i 295 | . . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
14 | 13 | notbii 312 | . . 3 ⊢ (¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
15 | 2, 14 | bitr2i 268 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) |
16 | 1, 15 | syl6ib 243 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 880 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3799 ⊊ wpss 3800 class class class wbr 4874 Cℋ cch 28342 ⋖ℋ ccv 28377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-cv 29694 |
This theorem is referenced by: cvmdi 29739 |
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