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Mirrors > Home > HSE Home > Th. List > cvnref | Structured version Visualization version GIF version |
Description: The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvnref | ⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvnsym 29149 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴)) | |
2 | 1 | anidms 677 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴)) |
3 | 2 | pm2.01d 181 | 1 ⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1990 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: (None) |
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