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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 30066 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 404 | . . 3 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵)) | |
3 | anass 471 | . . . . 5 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) | |
4 | dfpss2 4065 | . . . . . 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 280 | . . . 4 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
7 | 6 | notbii 322 | . . 3 ⊢ (¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
8 | 2, 7 | bitr2i 278 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)) |
9 | 1, 8 | syl6ib 253 | 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 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ⊊ wpss 3940 class class class wbr 5069 Cℋ cch 28709 ⋖ℋ ccv 28744 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-cv 30059 |
This theorem is referenced by: cvati 30146 cvexchlem 30148 atexch 30161 |
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