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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 29848 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴)) | |
2 | 1 | anidms 559 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴)) |
3 | 2 | pm2.01d 182 | 1 ⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2050 class class class wbr 4929 Cℋ cch 28485 ⋖ℋ ccv 28520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-cv 29837 |
This theorem is referenced by: (None) |
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