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Mirrors > Home > HSE Home > Th. List > cvnsym | Structured version Visualization version GIF version |
Description: The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvnsym | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvpss 30068 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵)) | |
2 | cvpss 30068 | . . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴)) | |
3 | 2 | ancoms 462 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴)) |
4 | pssn2lp 4029 | . . . . 5 ⊢ ¬ (𝐵 ⊊ 𝐴 ∧ 𝐴 ⊊ 𝐵) | |
5 | 4 | imnani 404 | . . . 4 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊊ 𝐵) |
6 | 3, 5 | syl6 35 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → ¬ 𝐴 ⊊ 𝐵)) |
7 | 6 | con2d 136 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴)) |
8 | 1, 7 | syld 47 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ⊊ wpss 3882 class class class wbr 5030 Cℋ cch 28712 ⋖ℋ ccv 28747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cv 30062 |
This theorem is referenced by: cvnref 30074 |
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