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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 29991 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 402 | . . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) | |
3 | eqcom 2828 | . . . 4 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
4 | 3 | imbi2i 337 | . . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶)) |
5 | dfpss2 4061 | . . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
6 | 5 | anbi1i 623 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵)) |
7 | an32 642 | . . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) | |
8 | 6, 7 | bitri 276 | . . . 4 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) |
9 | 8 | notbii 321 | . . 3 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶)) |
10 | 2, 4, 9 | 3bitr4ri 305 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)) |
11 | 1, 10 | syl6ib 252 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ⊊ wpss 3936 class class class wbr 5058 Cℋ cch 28634 ⋖ℋ ccv 28669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-cv 29984 |
This theorem is referenced by: atcveq0 30053 atcvatlem 30090 |
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