Theorem lhpex2leN 35299

Description: There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lhp2at.l	|- .<_ = ( le ` K )
lhp2at.a	|- A = ( Atoms ` K )
lhp2at.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
lhpex2leN	|- ( ( K e. HL /\ W e. H ) -> E. p e. A E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) )

Distinct variable groups:   q, p, A   H, p, q   K, p, q   .<_ , p, q   W, p, q

Proof of Theorem lhpex2leN

Step	Hyp	Ref	Expression
1		simprr 796	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> p .<_ W )
2		lhp2at.l	. . . . . 6 |- .<_ = ( le ` K )
3		lhp2at.a	. . . . . 6 |- A = ( Atoms ` K )
4		lhp2at.h	. . . . . 6 |- H = ( LHyp ` K )
5	2, 3, 4	lhpexle1 35294	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> E. q e. A ( q .<_ W /\ q =/= p ) )
6	5	adantr 481	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> E. q e. A ( q .<_ W /\ q =/= p ) )
7	1, 6	jca 554	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) )
8		necom 2847	. . . . . . 7 |- ( p =/= q <-> q =/= p )
9	8	3anbi3i 1255	. . . . . 6 |- ( ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> ( p .<_ W /\ q .<_ W /\ q =/= p ) )
10		3anass 1042	. . . . . 6 |- ( ( p .<_ W /\ q .<_ W /\ q =/= p ) <-> ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) )
11	9, 10	bitri 264	. . . . 5 |- ( ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) )
12	11	rexbii 3041	. . . 4 |- ( E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> E. q e. A ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) )
13		r19.42v 3092	. . . 4 |- ( E. q e. A ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) <-> ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) )
14	12, 13	bitr2i 265	. . 3 |- ( ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) <-> E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) )
15	7, 14	sylib 208	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) )
16	2, 3, 4	lhpexle 35291	. 2 |- ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W )
17	15, 16	reximddv 3018	1 |- ( ( K e. HL /\ W e. H ) -> E. p e. A E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270

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-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by: (None)