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 e. CH /\ B e. CH ) -> ( A <oH B <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. 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 e. CH <-> A e. CH ) )
2	1	anbi1d 741	. . . 4 |- ( y = A -> ( ( y e. CH /\ z e. CH ) <-> ( A e. CH /\ z e. CH ) ) )
3		psseq1 3694	. . . . 5 |- ( y = A -> ( y C. z <-> A C. z ) )
4		psseq1 3694	. . . . . . . 8 |- ( y = A -> ( y C. x <-> A C. x ) )
5	4	anbi1d 741	. . . . . . 7 |- ( y = A -> ( ( y C. x /\ x C. z ) <-> ( A C. x /\ x C. z ) ) )
6	5	rexbidv 3052	. . . . . 6 |- ( y = A -> ( E. x e. CH ( y C. x /\ x C. z ) <-> E. x e. CH ( A C. x /\ x C. z ) ) )
7	6	notbid 308	. . . . 5 |- ( y = A -> ( -. E. x e. CH ( y C. x /\ x C. z ) <-> -. E. x e. CH ( A C. x /\ x C. z ) ) )
8	3, 7	anbi12d 747	. . . 4 |- ( y = A -> ( ( y C. z /\ -. E. x e. CH ( y C. x /\ x C. z ) ) <-> ( A C. z /\ -. E. x e. CH ( A C. x /\ x C. z ) ) ) )
9	2, 8	anbi12d 747	. . 3 |- ( y = A -> ( ( ( y e. CH /\ z e. CH ) /\ ( y C. z /\ -. E. x e. CH ( y C. x /\ x C. z ) ) ) <-> ( ( A e. CH /\ z e. CH ) /\ ( A C. z /\ -. E. x e. CH ( A C. x /\ x C. z ) ) ) ) )
10		eleq1 2689	. . . . 5 |- ( z = B -> ( z e. CH <-> B e. CH ) )
11	10	anbi2d 740	. . . 4 |- ( z = B -> ( ( A e. CH /\ z e. CH ) <-> ( A e. CH /\ B e. CH ) ) )
12		psseq2 3695	. . . . 5 |- ( z = B -> ( A C. z <-> A C. B ) )
13		psseq2 3695	. . . . . . . 8 |- ( z = B -> ( x C. z <-> x C. B ) )
14	13	anbi2d 740	. . . . . . 7 |- ( z = B -> ( ( A C. x /\ x C. z ) <-> ( A C. x /\ x C. B ) ) )
15	14	rexbidv 3052	. . . . . 6 |- ( z = B -> ( E. x e. CH ( A C. x /\ x C. z ) <-> E. x e. CH ( A C. x /\ x C. B ) ) )
16	15	notbid 308	. . . . 5 |- ( z = B -> ( -. E. x e. CH ( A C. x /\ x C. z ) <-> -. E. x e. CH ( A C. x /\ x C. B ) ) )
17	12, 16	anbi12d 747	. . . 4 |- ( z = B -> ( ( A C. z /\ -. E. x e. CH ( A C. x /\ x C. z ) ) <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) )
18	11, 17	anbi12d 747	. . 3 |- ( z = B -> ( ( ( A e. CH /\ z e. CH ) /\ ( A C. z /\ -. E. x e. CH ( A C. x /\ x C. z ) ) ) <-> ( ( A e. CH /\ B e. CH ) /\ ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) ) )
19		df-cv 29138	. . 3 |- <oH = { <. y , z >. | ( ( y e. CH /\ z e. CH ) /\ ( y C. z /\ -. E. x e. CH ( y C. x /\ x C. z ) ) ) }
20	9, 18, 19	brabg 4994	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B <-> ( ( A e. CH /\ B e. CH ) /\ ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) ) )
21	20	bianabs 924	1 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C. wpss 3575   class class class wbr 4653   CHcch 27786   <oH 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