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Theorem isipodrs 15336
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 2443 . . . . 5  |-  ( Base `  (toInc `  A )
)  =  ( Base `  (toInc `  A )
)
21drsbn0 15112 . . . 4  |-  ( (toInc `  A )  e. Dirset  ->  (
Base `  (toInc `  A
) )  =/=  (/) )
32neneqd 2629 . . 3  |-  ( (toInc `  A )  e. Dirset  ->  -.  ( Base `  (toInc `  A ) )  =  (/) )
4 fvprc 5690 . . . . 5  |-  ( -.  A  e.  _V  ->  (toInc `  A )  =  (/) )
54fveq2d 5700 . . . 4  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (
Base `  (/) ) )
6 base0 14218 . . . 4  |-  (/)  =  (
Base `  (/) )
75, 6syl6eqr 2493 . . 3  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (/) )
83, 7nsyl2 127 . 2  |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
9 simp1 988 . 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 2443 . . . 4  |-  ( le
`  (toInc `  A
) )  =  ( le `  (toInc `  A ) )
111, 10isdrs 15109 . . 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 2443 . . . . . . . 8  |-  (toInc `  A )  =  (toInc `  A )
1312ipopos 15335 . . . . . . 7  |-  (toInc `  A )  e.  Poset
14 posprs 15124 . . . . . . 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 15330 . . . . . . 7  |-  ( A  e.  _V  ->  A  =  ( Base `  (toInc `  A ) ) )
19 neeq1 2621 . . . . . . . 8  |-  ( A  =  ( Base `  (toInc `  A ) )  -> 
( A  =/=  (/)  <->  ( Base `  (toInc `  A )
)  =/=  (/) ) )
20 rexeq 2923 . . . . . . . . . 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 2930 . . . . . . . . 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 2930 . . . . . . . 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 15333 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  x  e.  A  /\  z  e.  A )  ->  ( x ( le
`  (toInc `  A
) ) z  <->  x  C_  z
) )
2925, 26, 27, 28syl3anc 1218 . . . . . . . . . . 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 15333 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  y  e.  A  /\  z  e.  A )  ->  ( y ( le
`  (toInc `  A
) ) z  <->  y  C_  z ) )
3225, 30, 27, 31syl3anc 1218 . . . . . . . . . . 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 3535 . . . . . . . . . 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 2737 . . . . . . . 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 2760 . . . . . . 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 969 . . . 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 969 . . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    u. cun 3331    C_ wss 3333   (/)c0 3642   class class class wbr 4297   ` cfv 5423   Basecbs 14179   lecple 14250    Preset cpreset 15101  Dirsetcdrs 15102   Posetcpo 15115  toInccipo 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-tset 14262  df-ple 14263  df-ocomp 14264  df-preset 15103  df-drs 15104  df-poset 15121  df-ipo 15327
This theorem is referenced by:  ipodrscl  15337  fpwipodrs  15339  ipodrsima  15340  nacsfix  29053
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