Theorem cvbr 11854

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
cvbr	|- ((A e. CH /\ B e. CH) -> (A <o 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

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 679	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		psseq1 2697	. . . . 5 |- (y = A -> (y C. z <-> A C. z))
4		psseq1 2697	. . . . . . . 8 |- (y = A -> (y C. x <-> A C. x))
5	4	anbi1d 679	. . . . . . 7 |- (y = A -> ((y C. x /\ x C. z) <-> (A C. x /\ x C. z)))
6	5	rexbidv 2124	. . . . . 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 673	. . . . 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 690	. . . 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 690	. . 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 1957	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
11	10	anbi2d 678	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
12		psseq2 2698	. . . . 5 |- (z = B -> (A C. z <-> A C. B))
13		psseq2 2698	. . . . . . . 8 |- (z = B -> (x C. z <-> x C. B))
14	13	anbi2d 678	. . . . . . 7 |- (z = B -> ((A C. x /\ x C. z) <-> (A C. x /\ x C. B)))
15	14	rexbidv 2124	. . . . . 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 673	. . . . 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 690	. . . 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 690	. . 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 11851	. . 3 |- <o = {<.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 3568	. 2 |- ((A e. CH /\ B e. CH) -> (A <o B <-> ((A e. CH /\ B e. CH) /\ (A C. B /\ -. E.x e. CH (A C. x /\ x C. B)))))
21	20	bianabs 715	1 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A C. B /\ -. E.x e. CH (A C. x /\ x C. B))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   C. wpss 2594   class class class wbr 3338  CHcch 10430   <o ccv 10466

This theorem is referenced by:  cvbr2 11855  cvcon3 11856  cvpss 11857  cvnbtwn 11858

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-cv 11851

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