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Mirrors > Home > HSE Home > Th. List > cvnbtwn | 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 |
---|---|
cvnbtwn | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvbr 31802 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))) | |
2 | psseq2 4087 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝐶)) | |
3 | psseq1 4086 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵)) | |
4 | 2, 3 | anbi12d 629 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) |
5 | 4 | rspcev 3611 | . . . . . . 7 ⊢ ((𝐶 ∈ Cℋ ∧ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) |
6 | 5 | ex 411 | . . . . . 6 ⊢ (𝐶 ∈ Cℋ → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))) |
7 | 6 | con3rr3 155 | . . . . 5 ⊢ (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) |
8 | 7 | adantl 480 | . . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) |
9 | 1, 8 | syl6bi 252 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))) |
10 | 9 | com23 86 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐶 ∈ Cℋ → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))) |
11 | 10 | 3impia 1115 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 ⊊ wpss 3948 class class class wbr 5147 Cℋ cch 30449 ⋖ℋ ccv 30484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-cv 31799 |
This theorem is referenced by: cvnbtwn2 31807 cvnbtwn3 31808 cvnbtwn4 31809 cvntr 31812 |
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