Theorem cvbr 32306

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 2832	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 630	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		psseq1 4113	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝑧 ↔ 𝐴 ⊊ 𝑧))
4		psseq1 4113	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝑥))
5	4	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
6	5	rexbidv 3185	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
7	6	notbid 318	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))
8	3, 7	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)) ↔ (𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧))))
9	2, 8	anbi12d 631	. . 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 2832	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
11	10	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
12		psseq2 4114	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 ⊊ 𝑧 ↔ 𝐴 ⊊ 𝐵))
13		psseq2 4114	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑥 ⊊ 𝑧 ↔ 𝑥 ⊊ 𝐵))
14	13	anbi2d 629	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
15	14	rexbidv 3185	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
16	15	notbid 318	. . . . 5 ⊢ (𝑧 = 𝐵 → (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
17	12, 16	anbi12d 631	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
18	11, 17	anbi12d 631	. . 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 32303	. . 3 ⊢ ⋖ℋ = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ (𝑦 ⊊ 𝑧 ∧ ¬ ∃𝑥 ∈ Cℋ (𝑦 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝑧)))}
20	9, 18, 19	brabg 5558	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))))
21	20	bianabs 541	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ⊊ wpss 3977   class class class wbr 5166   C_ℋ cch 30953   ⋖_ℋ ccv 30988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cv 32303

This theorem is referenced by:  cvbr2  32307  cvcon3  32308  cvpss  32309  cvnbtwn  32310