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| Mirrors > Home > HSE Home > Th. List > cvbr2 | Structured version Visualization version GIF version | ||
| Description: Binary relation expressing 𝐵 covers 𝐴. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cvbr2 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cvbr 32301 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))) | |
| 2 | iman 401 | . . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵)) | |
| 3 | anass 468 | . . . . . . 7 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))) | |
| 4 | dfpss2 4088 | . . . . . . . 8 ⊢ (𝑥 ⊊ 𝐵 ↔ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)) | |
| 5 | 4 | anbi2i 623 | . . . . . . 7 ⊢ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))) |
| 6 | 3, 5 | bitr4i 278 | . . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) |
| 7 | 2, 6 | xchbinx 334 | . . . . 5 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) |
| 8 | 7 | ralbii 3093 | . . . 4 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) |
| 9 | ralnex 3072 | . . . 4 ⊢ (∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) | |
| 10 | 8, 9 | bitri 275 | . . 3 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) |
| 11 | 10 | anbi2i 623 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))) |
| 12 | 1, 11 | bitr4di 289 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ⊊ wpss 3952 class class class wbr 5143 Cℋ cch 30948 ⋖ℋ ccv 30983 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cv 32298 |
| This theorem is referenced by: spansncv2 32312 elat2 32359 |
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