Theorem cvbrt 10209

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 groups:   x,A   x,B

Proof of Theorem cvbrt

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 617	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		psseq1 2135	. . . . 5 |- (y = A -> (y (. z <-> A (. z))
4		psseq1 2135	. . . . . . . 8 |- (y = A -> (y (. x <-> A (. x))
5	4	anbi1d 617	. . . . . . 7 |- (y = A -> ((y (. x /\ x (. z) <-> (A (. x /\ x (. z)))
6	5	rexbidv 1664	. . . . . 6 |- (y = A -> (E.x e. CH (y (. x /\ x (. z) <-> E.x e. CH (A (. x /\ x (. z)))
7	6	negbid 611	. . . . 5 |- (y = A -> (-. E.x e. CH (y (. x /\ x (. z) <-> -. E.x e. CH (A (. x /\ x (. z)))
8	3, 7	anbi12d 628	. . . 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 628	. . 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 1534	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
11	10	anbi2d 616	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
12		psseq2 2136	. . . . 5 |- (z = B -> (A (. z <-> A (. B))
13		psseq2 2136	. . . . . . . 8 |- (z = B -> (x (. z <-> x (. B))
14	13	anbi2d 616	. . . . . . 7 |- (z = B -> ((A (. x /\ x (. z) <-> (A (. x /\ x (. B)))
15	14	rexbidv 1664	. . . . . 6 |- (z = B -> (E.x e. CH (A (. x /\ x (. z) <-> E.x e. CH (A (. x /\ x (. B)))
16	15	negbid 611	. . . . 5 |- (z = B -> (-. E.x e. CH (A (. x /\ x (. z) <-> -. E.x e. CH (A (. x /\ x (. B)))
17	12, 16	anbi12d 628	. . . 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 628	. . 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 10206	. . 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 2818	. 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	20	bianabs 653	1 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646   (. wpss 2048   class class class wbr 2619  CHcch 8798   <o ccv 8834

This theorem is referenced by:  cvbr2t 10210  cvcon3t 10211  cvpsst 10212  cvnbtwnt 10213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cv 10206

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