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Theorem r1wunlim 8617
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 449 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e. WUni
)
21wun0 8598 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  (/)  e.  ( R1 `  A ) )
3 elfvdm 5760 . . . . . 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 7696 . . . . . 6  |-  R1  Fn  On
6 fndm 5547 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
75, 6ax-mp 5 . . . . 5  |-  dom  R1  =  On
84, 7syl6eleq 2528 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  On )
9 eloni 4594 . . . 4  |-  ( A  e.  On  ->  Ord  A )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Ord  A )
11 n0i 3635 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  -.  ( R1 `  A )  =  (/) )
122, 11syl 16 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  =  (/) )
13 fveq2 5731 . . . . . 6  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
14 r10 7697 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1513, 14syl6eq 2486 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
1612, 15nsyl 116 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  A  =  (/) )
17 suceloni 4796 . . . . . . . 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 4662 . . . . . . . 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 7709 . . . . . . 7  |-  ( suc 
A  e.  On  ->  ( A  e.  suc  A  ->  ( R1 `  A
)  e.  ( R1
`  suc  A )
) )
2218, 20, 21sylc 59 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
23 r1elwf 7725 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
24 wfelirr 7754 . . . . . 6  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  ->  -.  ( R1 `  A
)  e.  ( R1
`  A ) )
2522, 23, 243syl 19 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  e.  ( R1 `  A ) )
26 simprr 735 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  =  suc  x )
2726fveq2d 5735 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
28 r1suc 7699 . . . . . . . . 9  |-  ( x  e.  On  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
2928ad2antrl 710 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
3027, 29eqtrd 2470 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 simplr 733 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e. WUni )
328adantr 453 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  e.  On )
33 sucidg 4662 . . . . . . . . . . 11  |-  ( x  e.  On  ->  x  e.  suc  x )
3433ad2antrl 710 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  suc  x )
3534, 26eleqtrrd 2515 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  A )
36 r1ord 7709 . . . . . . . . 9  |-  ( A  e.  On  ->  (
x  e.  A  -> 
( R1 `  x
)  e.  ( R1
`  A ) ) )
3732, 35, 36sylc 59 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  x )  e.  ( R1 `  A ) )
3831, 37wunpw 8587 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ~P ( R1 `  x )  e.  ( R1 `  A
) )
3930, 38eqeltrd 2512 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e.  ( R1 `  A ) )
4039rexlimdvaa 2833 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  e.  ( R1
`  A ) ) )
4125, 40mtod 171 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  E. x  e.  On  A  =  suc  x )
42 ioran 478 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
4316, 41, 42sylanbrc 647 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) )
44 dflim3 4830 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
4510, 43, 44sylanbrc 647 . 2  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Lim  A )
46 r1limwun 8616 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni
)
4745, 46impbida 807 1  |-  ( A  e.  V  ->  (
( R1 `  A
)  e. WUni  <->  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   Ord word 4583   Oncon0 4584   Lim wlim 4585   suc csuc 4586   dom cdm 4881   "cima 4884    Fn wfn 5452   ` cfv 5457   R1cr1 7691  WUnicwun 8580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-rdg 6671  df-r1 7693  df-rank 7694  df-wun 8582
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