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Theorem isipodrs 16005
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 2400 . . . . 5  |-  ( Base `  (toInc `  A )
)  =  ( Base `  (toInc `  A )
)
21drsbn0 15780 . . . 4  |-  ( (toInc `  A )  e. Dirset  ->  (
Base `  (toInc `  A
) )  =/=  (/) )
32neneqd 2603 . . 3  |-  ( (toInc `  A )  e. Dirset  ->  -.  ( Base `  (toInc `  A ) )  =  (/) )
4 fvprc 5797 . . . . 5  |-  ( -.  A  e.  _V  ->  (toInc `  A )  =  (/) )
54fveq2d 5807 . . . 4  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (
Base `  (/) ) )
6 base0 14772 . . . 4  |-  (/)  =  (
Base `  (/) )
75, 6syl6eqr 2459 . . 3  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (/) )
83, 7nsyl2 127 . 2  |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
9 simp1 995 . 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 2400 . . . 4  |-  ( le
`  (toInc `  A
) )  =  ( le `  (toInc `  A ) )
111, 10isdrs 15777 . . 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 2400 . . . . . . . 8  |-  (toInc `  A )  =  (toInc `  A )
1312ipopos 16004 . . . . . . 7  |-  (toInc `  A )  e.  Poset
14 posprs 15792 . . . . . . 7  |-  ( (toInc `  A )  e.  Poset  -> 
(toInc `  A )  e.  Preset  )
1513, 14mp1i 13 . . . . . 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 15999 . . . . . . 7  |-  ( A  e.  _V  ->  A  =  ( Base `  (toInc `  A ) ) )
19 neeq1 2682 . . . . . . . 8  |-  ( A  =  ( Base `  (toInc `  A ) )  -> 
( A  =/=  (/)  <->  ( Base `  (toInc `  A )
)  =/=  (/) ) )
20 rexeq 3002 . . . . . . . . . 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 3009 . . . . . . . . 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 3009 . . . . . . . 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 709 . . . . . . 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 17 . . . . . 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 752 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  A  e.  _V )
26 simplrl 760 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
27 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  z  e.  A )
2812, 10ipole 16002 . . . . . . . . . . . 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 761 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
3112, 10ipole 16002 . . . . . . . . . . . 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 709 . . . . . . . . . 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 3614 . . . . . . . . . 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 2912 . . . . . . . 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 2843 . . . . . . 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 702 . . . . . 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 709 . . . 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 976 . . . 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 976 . . . 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 351 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   A.wral 2751   E.wrex 2752   _Vcvv 3056    u. cun 3409    C_ wss 3411   (/)c0 3735   class class class wbr 4392   ` cfv 5523   Basecbs 14731   lecple 14806    Preset cpreset 15769  Dirsetcdrs 15770   Posetcpo 15783  toInccipo 15995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-tset 14818  df-ple 14819  df-ocomp 14820  df-preset 15771  df-drs 15772  df-poset 15789  df-ipo 15996
This theorem is referenced by:  ipodrscl  16006  fpwipodrs  16008  ipodrsima  16009  nacsfix  34970
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