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 e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. 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 e. CH <-> A e. CH))
2	1	anbi1d 469	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. 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 -> (E.x e. CH (y (. x /\ x (. z) <-> E.x e. CH (A (. x /\ x (. z)))
7	6	negbid 463	. . . . 5 |- (y = A -> (-. E.x e. CH (y (. x /\ x (. z) <-> -. E.x e. CH (A (. x /\ x (. z)))
8	3, 7	anbi12d 476	. . . 4 |- (y = A -> ((y (. z /\ -. E.x e. CH (y (. x /\ x (. z)) <-> (A (. z /\ -. E.x e. CH (A (. x /\ x (. z))))
9	2, 8	anbi12d 476	. . 3 |- (y = A -> (((y e. CH /\ z e. CH) /\ (y (. z /\ -. E.x e. CH (y (. x /\ x (. z))) <-> ((A e. CH /\ z e. CH) /\ (A (. z /\ -. E.x e. CH (A (. x /\ x (. z)))))
10		eleq1 1149	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
11	10	anbi2d 468	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. 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 -> (E.x e. CH (A (. x /\ x (. z) <-> E.x e. CH (A (. x /\ x (. B)))
16	15	negbid 463	. . . . 5 |- (z = B -> (-. E.x e. CH (A (. x /\ x (. z) <-> -. E.x e. CH (A (. x /\ x (. B)))
17	12, 16	anbi12d 476	. . . 4 |- (z = B -> ((A (. z /\ -. E.x e. CH (A (. x /\ x (. z)) <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))
18	11, 17	anbi12d 476	. . 3 |- (z = B -> (((A e. CH /\ z e. CH) /\ (A (. z /\ -. E.x e. CH (A (. x /\ x (. z))) <-> ((A e. CH /\ B e. CH) /\ (A (. B /\ -. E.x e. CH (A (. x /\ x (. B)))))
19		df-cv 5712	. . 3 |- <o = {<.y, z>. | ((y e. CH /\ z e. CH) /\ (y (. z /\ -. E.x e. CH (y (. x /\ x (. z)))}
20	9, 18, 19	brabg 2116	. 2 |- ((A e. CH /\ B e. CH) -> (A <o B <-> ((A e. CH /\ B e. CH) /\ (A (. B /\ -. E.x e. CH (A (. x /\ x (. B)))))
21		ibar 487	. 2 |- ((A e. CH /\ B e. CH) -> ((A (. B /\ -. E.x e. CH (A (. x /\ x (. B)) <-> ((A e. CH /\ B e. CH) /\ (A (. B /\ -. E.x e. CH (A (. x /\ x (. B)))))
22	20, 21	bitr4d 409	1 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202   (. wpss 1488   class class class wbr 2054  CHcch 4968   <o 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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