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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 29847 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 393 | . . 3 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵)) | |
3 | anass 461 | . . . . 5 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) | |
4 | dfpss2 3954 | . . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)) | |
5 | 4 | anbi2i 613 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) |
6 | 3, 5 | bitr4i 270 | . . . 4 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
7 | 6 | notbii 312 | . . 3 ⊢ (¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
8 | 2, 7 | bitr2i 268 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)) |
9 | 1, 8 | syl6ib 243 | 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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 ⊊ wpss 3832 class class class wbr 4930 Cℋ cch 28488 ⋖ℋ ccv 28523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 df-opab 4993 df-cv 29840 |
This theorem is referenced by: cvati 29927 cvexchlem 29929 atexch 29942 |
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