Theorem isipodrs 15665

Description: Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Assertion

Ref	Expression
isipodrs	|- ( (toInc ` A ) e. Dirset <-> ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) )

Distinct variable group:   z, A, x, y

Proof of Theorem isipodrs

Step	Hyp	Ref	Expression
1		eqid 2467	. . . . 5 |- ( Base ` (toInc ` A ) ) = ( Base ` (toInc ` A ) )
2	1	drsbn0 15441	. . . 4 |- ( (toInc ` A ) e. Dirset -> ( Base ` (toInc ` A ) ) =/= (/) )
3	2	neneqd 2669	. . 3 |- ( (toInc ` A ) e. Dirset -> -. ( Base ` (toInc ` A ) ) = (/) )
4		fvprc 5866	. . . . 5 |- ( -. A e. _V -> (toInc ` A ) = (/) )
5	4	fveq2d 5876	. . . 4 |- ( -. A e. _V -> ( Base ` (toInc ` A ) ) = ( Base ` (/) ) )
6		base0 14546	. . . 4 |- (/) = ( Base ` (/) )
7	5, 6	syl6eqr 2526	. . 3 |- ( -. A e. _V -> ( Base ` (toInc ` A ) ) = (/) )
8	3, 7	nsyl2 127	. 2 |- ( (toInc ` A ) e. Dirset -> A e. _V )
9		simp1 996	. 2 |- ( ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) -> A e. _V )
10		eqid 2467	. . . 4 |- ( le ` (toInc ` A ) ) = ( le ` (toInc ` A ) )
11	1, 10	isdrs 15438	. . 3 |- ( (toInc ` A ) e. Dirset <-> ( (toInc ` A ) e. Preset /\ ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) )
12		eqid 2467	. . . . . . . 8 |- (toInc ` A ) = (toInc ` A )
13	12	ipopos 15664	. . . . . . 7 |- (toInc ` A ) e. Poset
14		posprs 15453	. . . . . . 7 |- ( (toInc ` A ) e. Poset -> (toInc ` A ) e. Preset )
15	13, 14	mp1i 12	. . . . . 6 |- ( A e. _V -> (toInc ` A ) e. Preset )
16		id 22	. . . . . 6 |- ( A e. _V -> A e. _V )
17	15, 16	2thd 240	. . . . 5 |- ( A e. _V -> ( (toInc ` A ) e. Preset <-> A e. _V ) )
18	12	ipobas 15659	. . . . . . 7 |- ( A e. _V -> A = ( Base ` (toInc ` A ) ) )
19		neeq1 2748	. . . . . . . 8 |- ( A = ( Base ` (toInc ` A ) ) -> ( A =/= (/) <-> ( Base ` (toInc ` A ) ) =/= (/) ) )
20		rexeq 3064	. . . . . . . . . 10 |- ( A = ( Base ` (toInc ` A ) ) -> ( E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) )
21	20	raleqbi1dv 3071	. . . . . . . . 9 |- ( A = ( Base ` (toInc ` A ) ) -> ( A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) )
22	21	raleqbi1dv 3071	. . . . . . . 8 |- ( A = ( Base ` (toInc ` A ) ) -> ( A. x e. A A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) )
23	19, 22	anbi12d 710	. . . . . . 7 |- ( A = ( Base ` (toInc ` A ) ) -> ( ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) ) )
24	18, 23	syl 16	. . . . . 6 |- ( A e. _V -> ( ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) ) )
25		simpll 753	. . . . . . . . . . . 12 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> A e. _V )
26		simplrl 759	. . . . . . . . . . . 12 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> x e. A )
27		simpr 461	. . . . . . . . . . . 12 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> z e. A )
28	12, 10	ipole 15662	. . . . . . . . . . . 12 |- ( ( A e. _V /\ x e. A /\ z e. A ) -> ( x ( le ` (toInc ` A ) ) z <-> x C_ z ) )
29	25, 26, 27, 28	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( x ( le ` (toInc ` A ) ) z <-> x C_ z ) )
30		simplrr 760	. . . . . . . . . . . 12 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> y e. A )
31	12, 10	ipole 15662	. . . . . . . . . . . 12 |- ( ( A e. _V /\ y e. A /\ z e. A ) -> ( y ( le ` (toInc ` A ) ) z <-> y C_ z ) )
32	25, 30, 27, 31	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( y ( le ` (toInc ` A ) ) z <-> y C_ z ) )
33	29, 32	anbi12d 710	. . . . . . . . . 10 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> ( x C_ z /\ y C_ z ) ) )
34		unss 3683	. . . . . . . . . 10 |- ( ( x C_ z /\ y C_ z ) <-> ( x u. y ) C_ z )
35	33, 34	syl6bb 261	. . . . . . . . 9 |- ( ( ( A e. _V /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> ( x u. y ) C_ z ) )
36	35	rexbidva 2975	. . . . . . . 8 |- ( ( A e. _V /\ ( x e. A /\ y e. A ) ) -> ( E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> E. z e. A ( x u. y ) C_ z ) )
37	36	2ralbidva 2909	. . . . . . 7 |- ( A e. _V -> ( A. x e. A A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) <-> A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) )
38	37	anbi2d 703	. . . . . 6 |- ( A e. _V -> ( ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) )
39	24, 38	bitr3d 255	. . . . 5 |- ( A e. _V -> ( ( ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) )
40	17, 39	anbi12d 710	. . . 4 |- ( A e. _V -> ( ( (toInc ` A ) e. Preset /\ ( ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) ) <-> ( A e. _V /\ ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) ) )
41		3anass 977	. . . 4 |- ( ( (toInc ` A ) e. Preset /\ ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( (toInc ` A ) e. Preset /\ ( ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) ) )
42		3anass 977	. . . 4 |- ( ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) <-> ( A e. _V /\ ( A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) )
43	40, 41, 42	3bitr4g 288	. . 3 |- ( A e. _V -> ( ( (toInc ` A ) e. Preset /\ ( Base ` (toInc ` A ) ) =/= (/) /\ A. x e. ( Base ` (toInc ` A ) ) A. y e. ( Base ` (toInc ` A ) ) E. z e. ( Base ` (toInc ` A ) ) ( x ( le ` (toInc ` A ) ) z /\ y ( le ` (toInc ` A ) ) z ) ) <-> ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) )
44	11, 43	syl5bb 257	. 2 |- ( A e. _V -> ( (toInc ` A ) e. Dirset <-> ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) ) )
45	8, 9, 44	pm5.21nii 353	1 |- ( (toInc ` A ) e. Dirset <-> ( A e. _V /\ A =/= (/) /\ A. x e. A A. y e. A E. z e. A ( x u. y ) C_ z ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118   u. cun 3479   C_ wss 3481   (/)c0 3790   class class class wbr 4453   ` cfv 5594   Basecbs 14507   lecple 14579   Preset cpreset 15430  Dirsetcdrs 15431   Posetcpo 15444  toInccipo 15655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-tset 14591  df-ple 14592  df-ocomp 14593  df-preset 15432  df-drs 15433  df-poset 15450  df-ipo 15656

This theorem is referenced by:  ipodrscl  15666  fpwipodrs  15668  ipodrsima  15669  nacsfix  30572