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Theorem isipodrs 14264
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 2283 . . . . 5  |-  ( Base `  (toInc `  A )
)  =  ( Base `  (toInc `  A )
)
21drsbn0 14071 . . . 4  |-  ( (toInc `  A )  e. Dirset  ->  (
Base `  (toInc `  A
) )  =/=  (/) )
32neneqd 2462 . . 3  |-  ( (toInc `  A )  e. Dirset  ->  -.  ( Base `  (toInc `  A ) )  =  (/) )
4 fvprc 5519 . . . . 5  |-  ( -.  A  e.  _V  ->  (toInc `  A )  =  (/) )
54fveq2d 5529 . . . 4  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (
Base `  (/) ) )
6 base0 13185 . . . 4  |-  (/)  =  (
Base `  (/) )
75, 6syl6eqr 2333 . . 3  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (/) )
83, 7nsyl2 119 . 2  |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
9 simp1 955 . 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 2283 . . . 4  |-  ( le
`  (toInc `  A
) )  =  ( le `  (toInc `  A ) )
111, 10isdrs 14068 . . 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 2283 . . . . . . . 8  |-  (toInc `  A )  =  (toInc `  A )
1312ipopos 14263 . . . . . . 7  |-  (toInc `  A )  e.  Poset
14 posprs 14083 . . . . . . 7  |-  ( (toInc `  A )  e.  Poset  -> 
(toInc `  A )  e.  Preset  )
1513, 14mp1i 11 . . . . . 6  |-  ( A  e.  _V  ->  (toInc `  A )  e.  Preset  )
16 id 19 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  _V )
1715, 162thd 231 . . . . 5  |-  ( A  e.  _V  ->  (
(toInc `  A )  e.  Preset 
<->  A  e.  _V )
)
1812ipobas 14258 . . . . . . 7  |-  ( A  e.  _V  ->  A  =  ( Base `  (toInc `  A ) ) )
19 neeq1 2454 . . . . . . . 8  |-  ( A  =  ( Base `  (toInc `  A ) )  -> 
( A  =/=  (/)  <->  ( Base `  (toInc `  A )
)  =/=  (/) ) )
20 rexeq 2737 . . . . . . . . . 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 2744 . . . . . . . . 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 2744 . . . . . . . 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 691 . . . . . . 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 15 . . . . . 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 730 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  A  e.  _V )
26 simplrl 736 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
27 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  z  e.  A )
2812, 10ipole 14261 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  x  e.  A  /\  z  e.  A )  ->  ( x ( le
`  (toInc `  A
) ) z  <->  x  C_  z
) )
2925, 26, 27, 28syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
x ( le `  (toInc `  A ) ) z  <->  x  C_  z ) )
30 simplrr 737 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
3112, 10ipole 14261 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  y  e.  A  /\  z  e.  A )  ->  ( y ( le
`  (toInc `  A
) ) z  <->  y  C_  z ) )
3225, 30, 27, 31syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
y ( le `  (toInc `  A ) ) z  <->  y  C_  z
) )
3329, 32anbi12d 691 . . . . . . . . . 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 3349 . . . . . . . . . 10  |-  ( ( x  C_  z  /\  y  C_  z )  <->  ( x  u.  y )  C_  z
)
3533, 34syl6bb 252 . . . . . . . . 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 2560 . . . . . . . 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 2583 . . . . . . 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 684 . . . . . 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 246 . . . . 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 691 . . . 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 938 . . . 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 938 . . . 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 279 . . 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 248 . 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 342 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 set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215    Preset cpreset 14060  Dirsetcdrs 14061   Posetcpo 14074  toInccipo 14254
This theorem is referenced by:  ipodrscl  14265  fpwipodrs  14267  ipodrsima  14268  nacsfix  26787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-tset 13227  df-ple 13228  df-ocomp 13229  df-preset 14062  df-drs 14063  df-poset 14080  df-ipo 14255
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