Theorem cvbr2t 10148

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 10147	. 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 2129	. . . . . . . . 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 1664	. . . 4 |- (A.x e. CH ((A (. x /\ x (_ B) -> x = B) <-> A.x e. CH -. (A (. x /\ x (. B))
10		ralnex 1650	. . . 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 537	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 954   e. wcel 956  A.wral 1642  E.wrex 1643   (_ wss 2043   (. wpss 2044   class class class wbr 2614  CHcch 8737   <o ccv 8773

This theorem is referenced by:  spansncv2t 10158  elat2 10204

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cv 10144

Copyright terms: Public domain