Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 29145 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))) | |
2 | iman 440 | . . 3 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵)) | |
3 | anass 681 | . . . . 5 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) | |
4 | dfpss2 3692 | . . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)) | |
5 | 4 | anbi2i 730 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))) |
6 | 3, 5 | bitr4i 267 | . . . 4 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
7 | 6 | notbii 310 | . . 3 ⊢ (¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) |
8 | 2, 7 | bitr2i 265 | . 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)) |
9 | 1, 8 | syl6ib 241 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ⊊ wpss 3575 class class class wbr 4653 Cℋ cch 27786 ⋖ℋ ccv 27821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cv 29138 |
This theorem is referenced by: cvati 29225 cvexchlem 29227 atexch 29240 |
Copyright terms: Public domain | W3C validator |