Theorem cvbrt 5714

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.

Assertion

Ref	Expression
cvbrt	⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))))

Distinct variable group(s):   x,A   x,B

Proof of Theorem cvbrt

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (y = A → (y ∈ Cℋ ↔ A ∈ Cℋ ))
2	1	anbi1d 469	. . . 4 ⊢ (y = A → ((y ∈ Cℋ ∧ z ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ z ∈ Cℋ )))
3		psseq1 1559	. . . . 5 ⊢ (y = A → (y ⊂ z ↔ A ⊂ z))
4		psseq1 1559	. . . . . . . 8 ⊢ (y = A → (y ⊂ x ↔ A ⊂ x))
5	4	anbi1d 469	. . . . . . 7 ⊢ (y = A → ((y ⊂ x ∧ x ⊂ z) ↔ (A ⊂ x ∧ x ⊂ z)))
6	5	birexdv 1220	. . . . . 6 ⊢ (y = A → (∃x ∈ Cℋ (y ⊂ x ∧ x ⊂ z) ↔ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z)))
7	6	negbid 463	. . . . 5 ⊢ (y = A → (¬ ∃x ∈ Cℋ (y ⊂ x ∧ x ⊂ z) ↔ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z)))
8	3, 7	anbi12d 476	. . . 4 ⊢ (y = A → ((y ⊂ z ∧ ¬ ∃x ∈ Cℋ (y ⊂ x ∧ x ⊂ z)) ↔ (A ⊂ z ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z))))
9	2, 8	anbi12d 476	. . 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 1149	. . . . 5 ⊢ (z = B → (z ∈ Cℋ ↔ B ∈ Cℋ ))
11	10	anbi2d 468	. . . 4 ⊢ (z = B → ((A ∈ Cℋ ∧ z ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ B ∈ Cℋ )))
12		psseq2 1560	. . . . 5 ⊢ (z = B → (A ⊂ z ↔ A ⊂ B))
13		psseq2 1560	. . . . . . . 8 ⊢ (z = B → (x ⊂ z ↔ x ⊂ B))
14	13	anbi2d 468	. . . . . . 7 ⊢ (z = B → ((A ⊂ x ∧ x ⊂ z) ↔ (A ⊂ x ∧ x ⊂ B)))
15	14	birexdv 1220	. . . . . 6 ⊢ (z = B → (∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z) ↔ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))
16	15	negbid 463	. . . . 5 ⊢ (z = B → (¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z) ↔ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))
17	12, 16	anbi12d 476	. . . 4 ⊢ (z = B → ((A ⊂ z ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ z)) ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))))
18	11, 17	anbi12d 476	. . 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 5712	. . 3 ⊢ ⋖ = {⟨y, z⟩∣((y ∈ Cℋ ∧ z ∈ Cℋ ) ∧ (y ⊂ z ∧ ¬ ∃x ∈ Cℋ (y ⊂ x ∧ x ⊂ z)))}
20	9, 18, 19	brabg 2116	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ ((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))))
21		ibar 487	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → ((A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)) ↔ ((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))))
22	20, 21	bitr4d 409	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊂ wpss 1488   class class class wbr 2054   C_ℋ cch 4968   ⋖ ccv 4981

This theorem is referenced by:  cvbr2t 5715  cvcon3t 5716  cvpsst 5717  cvnbtwnt 5718

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cv 5712

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