Theorem cvbr 29141

Description: Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvbr	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cvbr

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 741	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		psseq1 3694	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝑧 ↔ 𝐴 ⊊ 𝑧))
4		psseq1 3694	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝑥))
5	4	anbi1d 741	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
6	5	rexbidv 3052	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
7	6	notbid 308	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
8	3, 7	anbi12d 747	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)) ↔ (𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧))))
9	2, 8	anbi12d 747	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ (𝑦 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧))) ↔ ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ (𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))))
10		eleq1 2689	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
11	10	anbi2d 740	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
12		psseq2 3695	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 ⊊ 𝑧 ↔ 𝐴 ⊊ 𝐵))
13		psseq2 3695	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑥 ⊊ 𝑧 ↔ 𝑥 ⊊ 𝐵))
14	13	anbi2d 740	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
15	14	rexbidv 3052	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
16	15	notbid 308	. . . . 5 ⊢ (𝑧 = 𝐵 → (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
17	12, 16	anbi12d 747	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
18	11, 17	anbi12d 747	. . 3 ⊢ (𝑧 = 𝐵 → (((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ (𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧))) ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))))
19		df-cv 29138	. . 3 ⊢ ⋖ℋ = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ (𝑦 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))}
20	9, 18, 19	brabg 4994	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))))
21	20	bianabs 924	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊊ wpss 3575   class class class wbr 4653   C_ℋ 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:  cvbr2  29142  cvcon3  29143  cvpss  29144  cvnbtwn  29145