Theorem cvbr2t 10210

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 10209	. 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 2133	. . . . . . . . 9 |- (x (. B <-> (x (_ B /\ -. x = B))
5	4	anbi2i 480	. . . . . . . 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 1667	. . . 4 |- (A.x e. CH ((A (. x /\ x (_ B) -> x = B) <-> A.x e. CH -. (A (. x /\ x (. B))
10		ralnex 1653	. . . 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 480	. 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 538	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 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047   (. wpss 2048   class class class wbr 2619  CHcch 8798   <o ccv 8834

This theorem is referenced by:  spansncv2t 10220  elat2 10267

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-ral 1649  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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