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Mirrors > Home > HSE Home > Th. List > cvnbtwn2 | 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 |
---|---|
cvnbtwn2 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvnbtwn 31270 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 403 | . . 3 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵)) | |
3 | anass 470 | . . . . 5 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) | |
4 | dfpss2 4050 | . . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)) | |
5 | 4 | anbi2i 624 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) |
6 | 3, 5 | bitr4i 278 | . . . 4 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
7 | 6 | notbii 320 | . . 3 ⊢ (¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
8 | 2, 7 | bitr2i 276 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)) |
9 | 1, 8 | syl6ib 251 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 ⊊ wpss 3916 class class class wbr 5110 Cℋ cch 29913 ⋖ℋ ccv 29948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-cv 31263 |
This theorem is referenced by: cvati 31350 cvexchlem 31352 atexch 31365 |
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