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Mirrors > Home > HSE Home > Th. List > cvnbtwn3 | Structured version Visualization version GIF version |
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvnbtwn3 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvnbtwn 32168 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 400 | . . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) | |
3 | eqcom 2732 | . . . 4 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
4 | 3 | imbi2i 335 | . . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶)) |
5 | dfpss2 4081 | . . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
6 | 5 | anbi1i 622 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵)) |
7 | an32 644 | . . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) | |
8 | 6, 7 | bitri 274 | . . . 4 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) |
9 | 8 | notbii 319 | . . 3 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) |
10 | 2, 4, 9 | 3bitr4ri 303 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)) |
11 | 1, 10 | imbitrdi 250 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ⊊ wpss 3945 class class class wbr 5149 Cℋ cch 30811 ⋖ℋ ccv 30846 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-cv 32161 |
This theorem is referenced by: atcveq0 32230 atcvatlem 32267 |
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