Theorem leat2 34581

Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)

Hypotheses

Ref	Expression
leatom.b	|- B = ( Base ` K )
leatom.l	|- .<_ = ( le ` K )
leatom.z	|- .0. = ( 0. ` K )
leatom.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
leat2	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ ( X =/= .0. /\ X .<_ P ) ) -> X = P )

Proof of Theorem leat2

Step	Hyp	Ref	Expression
1		leatom.b	. . . . . 6 |- B = ( Base ` K )
2		leatom.l	. . . . . 6 |- .<_ = ( le ` K )
3		leatom.z	. . . . . 6 |- .0. = ( 0. ` K )
4		leatom.a	. . . . . 6 |- A = ( Atoms ` K )
5	1, 2, 3, 4	leatb 34579	. . . . 5 |- ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X = P \/ X = .0. ) ) )
6		orcom 402	. . . . . 6 |- ( ( X = P \/ X = .0. ) <-> ( X = .0. \/ X = P ) )
7		neor 2885	. . . . . 6 |- ( ( X = .0. \/ X = P ) <-> ( X =/= .0. -> X = P ) )
8	6, 7	bitri 264	. . . . 5 |- ( ( X = P \/ X = .0. ) <-> ( X =/= .0. -> X = P ) )
9	5, 8	syl6bb 276	. . . 4 |- ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X =/= .0. -> X = P ) ) )
10	9	biimpd 219	. . 3 |- ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P -> ( X =/= .0. -> X = P ) ) )
11	10	com23 86	. 2 |- ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X =/= .0. -> ( X .<_ P -> X = P ) ) )
12	11	imp32 449	1 |- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ ( X =/= .0. /\ X .<_ P ) ) -> X = P )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   0.cp0 17037   OPcops 34459   Atomscatm 34550

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-preset 16928  df-poset 16946  df-plt 16958  df-glb 16975  df-p0 17039  df-oposet 34463  df-covers 34553  df-ats 34554

This theorem is referenced by:  dalemcea  34946  cdlemg12g  35937