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Theorem r1wunlim 8909
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 8890 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  (/)  e.  ( R1 `  A ) )
3 elfvdm 5721 . . . . . 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 7979 . . . . . 6  |-  R1  Fn  On
6 fndm 5515 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
75, 6ax-mp 5 . . . . 5  |-  dom  R1  =  On
84, 7syl6eleq 2533 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  On )
9 eloni 4734 . . . 4  |-  ( A  e.  On  ->  Ord  A )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Ord  A )
11 n0i 3647 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  -.  ( R1 `  A )  =  (/) )
122, 11syl 16 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  =  (/) )
13 fveq2 5696 . . . . . 6  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
14 r10 7980 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1513, 14syl6eq 2491 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
1612, 15nsyl 121 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  A  =  (/) )
17 suceloni 6429 . . . . . . . 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 4802 . . . . . . . 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 7992 . . . . . . 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 8008 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
24 wfelirr 8037 . . . . . 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 756 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  =  suc  x )
2726fveq2d 5700 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
28 r1suc 7982 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 simplr 754 . . . . . . . 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 4802 . . . . . . . . . . 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 2520 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  A )
36 r1ord 7992 . . . . . . . . 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 8879 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ~P ( R1 `  x )  e.  ( R1 `  A
) )
3930, 38eqeltrd 2517 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e.  ( R1 `  A ) )
4039rexlimdvaa 2847 . . . . 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 6463 . . 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 8908 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni
)
4745, 46impbida 828 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 1369    e. wcel 1756   E.wrex 2721   (/)c0 3642   ~Pcpw 3865   U.cuni 4096   Ord word 4723   Oncon0 4724   Lim wlim 4725   suc csuc 4726   dom cdm 4845   "cima 4848    Fn wfn 5418   ` cfv 5423   R1cr1 7974  WUnicwun 8872
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
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-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-om 6482  df-recs 6837  df-rdg 6871  df-r1 7976  df-rank 7977  df-wun 8874
This theorem is referenced by: (None)
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