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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
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  ( Base `  (toInc `  A )
)  =  ( Base `  (toInc `  A )
)
21drsbn0 15441 . . . 4  |-  ( (toInc `  A )  e. Dirset  ->  (
Base `  (toInc `  A
) )  =/=  (/) )
32neneqd 2669 . . 3  |-  ( (toInc `  A )  e. Dirset  ->  -.  ( Base `  (toInc `  A ) )  =  (/) )
4 fvprc 5866 . . . . 5  |-  ( -.  A  e.  _V  ->  (toInc `  A )  =  (/) )
54fveq2d 5876 . . . 4  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (
Base `  (/) ) )
6 base0 14546 . . . 4  |-  (/)  =  (
Base `  (/) )
75, 6syl6eqr 2526 . . 3  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (/) )
83, 7nsyl2 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 ) )
111, 10isdrs 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 )
1312ipopos 15664 . . . . . . 7  |-  (toInc `  A )  e.  Poset
14 posprs 15453 . . . . . . 7  |-  ( (toInc `  A )  e.  Poset  -> 
(toInc `  A )  e.  Preset  )
1513, 14mp1i 12 . . . . . 6  |-  ( A  e.  _V  ->  (toInc `  A )  e.  Preset  )
16 id 22 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  _V )
1715, 162thd 240 . . . . 5  |-  ( A  e.  _V  ->  (
(toInc `  A )  e.  Preset 
<->  A  e.  _V )
)
1812ipobas 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 ) ) )
2120raleqbi1dv 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 ) ) )
2221raleqbi1dv 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 ) ) )
2319, 22anbi12d 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 ) ) ) )
2418, 23syl 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 )
2812, 10ipole 15662 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  x  e.  A  /\  z  e.  A )  ->  ( x ( le
`  (toInc `  A
) ) z  <->  x  C_  z
) )
2925, 26, 27, 28syl3anc 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 )
3112, 10ipole 15662 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  y  e.  A  /\  z  e.  A )  ->  ( y ( le
`  (toInc `  A
) ) z  <->  y  C_  z ) )
3225, 30, 27, 31syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
y ( le `  (toInc `  A ) ) z  <->  y  C_  z
) )
3329, 32anbi12d 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
)
3533, 34syl6bb 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
) )
3635rexbidva 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
) )
37362ralbidva 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
) )
3837anbi2d 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
) ) )
3924, 38bitr3d 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
) ) )
4017, 39anbi12d 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
) ) )
4340, 41, 423bitr4g 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 )
) )
4411, 43syl5bb 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 ) ) )
458, 9, 44pm5.21nii 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
) )
Colors of variables: wff setvar class
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
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