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Theorem isipodrs 14579
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 2435 . . . . 5  |-  ( Base `  (toInc `  A )
)  =  ( Base `  (toInc `  A )
)
21drsbn0 14386 . . . 4  |-  ( (toInc `  A )  e. Dirset  ->  (
Base `  (toInc `  A
) )  =/=  (/) )
32neneqd 2614 . . 3  |-  ( (toInc `  A )  e. Dirset  ->  -.  ( Base `  (toInc `  A ) )  =  (/) )
4 fvprc 5714 . . . . 5  |-  ( -.  A  e.  _V  ->  (toInc `  A )  =  (/) )
54fveq2d 5724 . . . 4  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (
Base `  (/) ) )
6 base0 13498 . . . 4  |-  (/)  =  (
Base `  (/) )
75, 6syl6eqr 2485 . . 3  |-  ( -.  A  e.  _V  ->  (
Base `  (toInc `  A
) )  =  (/) )
83, 7nsyl2 121 . 2  |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
9 simp1 957 . 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 2435 . . . 4  |-  ( le
`  (toInc `  A
) )  =  ( le `  (toInc `  A ) )
111, 10isdrs 14383 . . 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 2435 . . . . . . . 8  |-  (toInc `  A )  =  (toInc `  A )
1312ipopos 14578 . . . . . . 7  |-  (toInc `  A )  e.  Poset
14 posprs 14398 . . . . . . 7  |-  ( (toInc `  A )  e.  Poset  -> 
(toInc `  A )  e.  Preset  )
1513, 14mp1i 12 . . . . . 6  |-  ( A  e.  _V  ->  (toInc `  A )  e.  Preset  )
16 id 20 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  _V )
1715, 162thd 232 . . . . 5  |-  ( A  e.  _V  ->  (
(toInc `  A )  e.  Preset 
<->  A  e.  _V )
)
1812ipobas 14573 . . . . . . 7  |-  ( A  e.  _V  ->  A  =  ( Base `  (toInc `  A ) ) )
19 neeq1 2606 . . . . . . . 8  |-  ( A  =  ( Base `  (toInc `  A ) )  -> 
( A  =/=  (/)  <->  ( Base `  (toInc `  A )
)  =/=  (/) ) )
20 rexeq 2897 . . . . . . . . . 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 2904 . . . . . . . . 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 2904 . . . . . . . 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 692 . . . . . . 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 731 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  A  e.  _V )
26 simplrl 737 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
27 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  z  e.  A )
2812, 10ipole 14576 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  x  e.  A  /\  z  e.  A )  ->  ( x ( le
`  (toInc `  A
) ) z  <->  x  C_  z
) )
2925, 26, 27, 28syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
x ( le `  (toInc `  A ) ) z  <->  x  C_  z ) )
30 simplrr 738 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
3112, 10ipole 14576 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  y  e.  A  /\  z  e.  A )  ->  ( y ( le
`  (toInc `  A
) ) z  <->  y  C_  z ) )
3225, 30, 27, 31syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
y ( le `  (toInc `  A ) ) z  <->  y  C_  z
) )
3329, 32anbi12d 692 . . . . . . . . . 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 3513 . . . . . . . . . 10  |-  ( ( x  C_  z  /\  y  C_  z )  <->  ( x  u.  y )  C_  z
)
3533, 34syl6bb 253 . . . . . . . . 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 2714 . . . . . . . 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 2737 . . . . . . 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 685 . . . . . 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 247 . . . . 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 692 . . . 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 940 . . . 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 940 . . . 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 280 . . 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 249 . 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 343 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    u. cun 3310    C_ wss 3312   (/)c0 3620   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528    Preset cpreset 14375  Dirsetcdrs 14376   Posetcpo 14389  toInccipo 14569
This theorem is referenced by:  ipodrscl  14580  fpwipodrs  14582  ipodrsima  14583  nacsfix  26757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-tset 13540  df-ple 13541  df-ocomp 13542  df-preset 14377  df-drs 14378  df-poset 14395  df-ipo 14570
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