Theorem opnoncon 34495

Description: Law of contradiction for orthoposets. (chocin 28354 analog.) (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
opnoncon.b	|- B = ( Base ` K )
opnoncon.o	|- ._|_ = ( oc ` K )
opnoncon.m	|- ./\ = ( meet ` K )
opnoncon.z	|- .0. = ( 0. ` K )

Assertion

Ref	Expression
opnoncon	|- ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )

Proof of Theorem opnoncon

Step	Hyp	Ref	Expression
1		opnoncon.b	. . . 4 |- B = ( Base ` K )
2		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
3		opnoncon.o	. . . 4 |- ._|_ = ( oc ` K )
4		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
5		opnoncon.m	. . . 4 |- ./\ = ( meet ` K )
6		opnoncon.z	. . . 4 |- .0. = ( 0. ` K )
7		eqid 2622	. . . 4 |- ( 1. ` K ) = ( 1. ` K )
8	1, 2, 3, 4, 5, 6, 7	oposlem 34469	. . 3 |- ( ( K e. OP /\ X e. B /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
9	8	3anidm23 1385	. 2 |- ( ( K e. OP /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
10	9	simp3d 1075	1 |- ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   joincjn 16944   meetcmee 16945   0.cp0 17037   1.cp1 17038   OPcops 34459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oposet 34463

This theorem is referenced by:  omlfh1N  34545  omlspjN  34548  atlatmstc  34606  pnonsingN  35219  lhpocnle  35302  dochnoncon  36680