Theorem cvbr2t 5715

Description: Binary relation expressing B covers A. Definition of covers in [Kalmbach] p. 15.

Assertion

Ref	Expression
cvbr2t	⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ∀x ∈ Cℋ ((A ⊂ x ∧ x ⊆ B) → x = B))))

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

Proof of Theorem cvbr2t

Step	Hyp	Ref	Expression
1		cvbrt 5714	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))))
2		iman 205	. . . . . 6 ⊢ (((A ⊂ x ∧ x ⊆ B) → x = B) ↔ ¬ ((A ⊂ x ∧ x ⊆ B) ∧ ¬ x = B))
3		anass 336	. . . . . . . 8 ⊢ (((A ⊂ x ∧ x ⊆ B) ∧ ¬ x = B) ↔ (A ⊂ x ∧ (x ⊆ B ∧ ¬ x = B)))
4		dfpss2 1557	. . . . . . . . 9 ⊢ (x ⊂ B ↔ (x ⊆ B ∧ ¬ x = B))
5	4	anbi2i 367	. . . . . . . 8 ⊢ ((A ⊂ x ∧ x ⊂ B) ↔ (A ⊂ x ∧ (x ⊆ B ∧ ¬ x = B)))
6	3, 5	bitr4 154	. . . . . . 7 ⊢ (((A ⊂ x ∧ x ⊆ B) ∧ ¬ x = B) ↔ (A ⊂ x ∧ x ⊂ B))
7	6	negbii 162	. . . . . 6 ⊢ (¬ ((A ⊂ x ∧ x ⊆ B) ∧ ¬ x = B) ↔ ¬ (A ⊂ x ∧ x ⊂ B))
8	2, 7	bitr 151	. . . . 5 ⊢ (((A ⊂ x ∧ x ⊆ B) → x = B) ↔ ¬ (A ⊂ x ∧ x ⊂ B))
9	8	biral 1223	. . . 4 ⊢ (∀x ∈ Cℋ ((A ⊂ x ∧ x ⊆ B) → x = B) ↔ ∀x ∈ Cℋ ¬ (A ⊂ x ∧ x ⊂ B))
10		ralnex 1209	. . . 4 ⊢ (∀x ∈ Cℋ ¬ (A ⊂ x ∧ x ⊂ B) ↔ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))
11	9, 10	bitr 151	. . 3 ⊢ (∀x ∈ Cℋ ((A ⊂ x ∧ x ⊆ B) → x = B) ↔ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))
12	11	anbi2i 367	. 2 ⊢ ((A ⊂ B ∧ ∀x ∈ Cℋ ((A ⊂ x ∧ x ⊆ B) → x = B)) ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))
13	1, 12	syl6bbr 416	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ∀x ∈ Cℋ ((A ⊂ x ∧ x ⊆ B) → x = B))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487   ⊂ wpss 1488   class class class wbr 2054   C_ℋ cch 4968   ⋖ ccv 4981

This theorem is referenced by:  spansncv2t 5725  elat2 5739

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-ral 1205  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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