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Theorem r1wunlim 9118
Description: The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
r1wunlim  |-  ( A  e.  V  ->  (
( R1 `  A
)  e. WUni  <->  Lim  A ) )

Proof of Theorem r1wunlim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e. WUni
)
21wun0 9099 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  (/)  e.  ( R1 `  A ) )
3 elfvdm 5882 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  A  e.  dom  R1 )
42, 3syl 16 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  dom  R1 )
5 r1fnon 8188 . . . . . 6  |-  R1  Fn  On
6 fndm 5670 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
75, 6ax-mp 5 . . . . 5  |-  dom  R1  =  On
84, 7syl6eleq 2541 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  On )
9 eloni 4878 . . . 4  |-  ( A  e.  On  ->  Ord  A )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Ord  A )
11 n0i 3775 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  -.  ( R1 `  A )  =  (/) )
122, 11syl 16 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  =  (/) )
13 fveq2 5856 . . . . . 6  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
14 r10 8189 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1513, 14syl6eq 2500 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
1612, 15nsyl 121 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  A  =  (/) )
17 suceloni 6633 . . . . . . . 8  |-  ( A  e.  On  ->  suc  A  e.  On )
188, 17syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  suc  A  e.  On )
19 sucidg 4946 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  suc  A )
208, 19syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  suc  A )
21 r1ord 8201 . . . . . . 7  |-  ( suc 
A  e.  On  ->  ( A  e.  suc  A  ->  ( R1 `  A
)  e.  ( R1
`  suc  A )
) )
2218, 20, 21sylc 60 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
23 r1elwf 8217 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
24 wfelirr 8246 . . . . . 6  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  ->  -.  ( R1 `  A
)  e.  ( R1
`  A ) )
2522, 23, 243syl 20 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  e.  ( R1 `  A ) )
26 simprr 757 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  =  suc  x )
2726fveq2d 5860 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
28 r1suc 8191 . . . . . . . . 9  |-  ( x  e.  On  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
2928ad2antrl 727 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
3027, 29eqtrd 2484 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 simplr 755 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e. WUni )
328adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  e.  On )
33 sucidg 4946 . . . . . . . . . . 11  |-  ( x  e.  On  ->  x  e.  suc  x )
3433ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  suc  x )
3534, 26eleqtrrd 2534 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  A )
36 r1ord 8201 . . . . . . . . 9  |-  ( A  e.  On  ->  (
x  e.  A  -> 
( R1 `  x
)  e.  ( R1
`  A ) ) )
3732, 35, 36sylc 60 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  x )  e.  ( R1 `  A ) )
3831, 37wunpw 9088 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ~P ( R1 `  x )  e.  ( R1 `  A
) )
3930, 38eqeltrd 2531 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e.  ( R1 `  A ) )
4039rexlimdvaa 2936 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  e.  ( R1
`  A ) ) )
4125, 40mtod 177 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  E. x  e.  On  A  =  suc  x )
42 ioran 490 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
4316, 41, 42sylanbrc 664 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) )
44 dflim3 6667 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
4510, 43, 44sylanbrc 664 . 2  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Lim  A )
46 r1limwun 9117 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni
)
4745, 46impbida 832 1  |-  ( A  e.  V  ->  (
( R1 `  A
)  e. WUni  <->  Lim  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   E.wrex 2794   (/)c0 3770   ~Pcpw 3997   U.cuni 4234   Ord word 4867   Oncon0 4868   Lim wlim 4869   suc csuc 4870   dom cdm 4989   "cima 4992    Fn wfn 5573   ` cfv 5578   R1cr1 8183  WUnicwun 9081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186  df-wun 9083
This theorem is referenced by: (None)
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