Theorem cvbr2t 10120

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

Assertion

Ref	Expression
cvbr2t	|- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ A.x e. CH ((A (. x /\ x (_ B) -> x = B))))

Distinct variable groups:   x,A   x,B

Proof of Theorem cvbr2t

Step	Hyp	Ref	Expression
1		cvbrt 10119	. 2 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))
2		iman 237	. . . . . 6 |- (((A (. x /\ x (_ B) -> x = B) <-> -. ((A (. x /\ x (_ B) /\ -. x = B))
3		anass 439	. . . . . . . 8 |- (((A (. x /\ x (_ B) /\ -. x = B) <-> (A (. x /\ (x (_ B /\ -. x = B)))
4		dfpss2 2123	. . . . . . . . 9 |- (x (. B <-> (x (_ B /\ -. x = B))
5	4	anbi2i 479	. . . . . . . 8 |- ((A (. x /\ x (. B) <-> (A (. x /\ (x (_ B /\ -. x = B)))
6	3, 5	bitr4 176	. . . . . . 7 |- (((A (. x /\ x (_ B) /\ -. x = B) <-> (A (. x /\ x (. B))
7	6	negbii 187	. . . . . 6 |- (-. ((A (. x /\ x (_ B) /\ -. x = B) <-> -. (A (. x /\ x (. B))
8	2, 7	bitr 173	. . . . 5 |- (((A (. x /\ x (_ B) -> x = B) <-> -. (A (. x /\ x (. B))
9	8	ralbii 1659	. . . 4 |- (A.x e. CH ((A (. x /\ x (_ B) -> x = B) <-> A.x e. CH -. (A (. x /\ x (. B))
10		ralnex 1645	. . . 4 |- (A.x e. CH -. (A (. x /\ x (. B) <-> -. E.x e. CH (A (. x /\ x (. B))
11	9, 10	bitr 173	. . 3 |- (A.x e. CH ((A (. x /\ x (_ B) -> x = B) <-> -. E.x e. CH (A (. x /\ x (. B))
12	11	anbi2i 479	. 2 |- ((A (. B /\ A.x e. CH ((A (. x /\ x (_ B) -> x = B)) <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B)))
13	1, 12	syl6bbr 536	1 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ A.x e. CH ((A (. x /\ x (_ B) -> x = B))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638   (_ wss 2037   (. wpss 2038   class class class wbr 2609  CHcch 8737   <o ccv 8773

This theorem is referenced by:  spansncv2t 10130  elat2 10175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-cv 10116

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