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Theorem r1wunlim 8568
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 448 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e. WUni
)
21wun0 8549 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  (/)  e.  ( R1 `  A ) )
3 elfvdm 5716 . . . . . 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 7649 . . . . . 6  |-  R1  Fn  On
6 fndm 5503 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
75, 6ax-mp 8 . . . . 5  |-  dom  R1  =  On
84, 7syl6eleq 2494 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  On )
9 eloni 4551 . . . 4  |-  ( A  e.  On  ->  Ord  A )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Ord  A )
11 n0i 3593 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  -.  ( R1 `  A )  =  (/) )
122, 11syl 16 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  =  (/) )
13 fveq2 5687 . . . . . 6  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
14 r10 7650 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1513, 14syl6eq 2452 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
1612, 15nsyl 115 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  A  =  (/) )
17 suceloni 4752 . . . . . . . 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 4619 . . . . . . . 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 7662 . . . . . . 7  |-  ( suc 
A  e.  On  ->  ( A  e.  suc  A  ->  ( R1 `  A
)  e.  ( R1
`  suc  A )
) )
2218, 20, 21sylc 58 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
23 r1elwf 7678 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
24 wfelirr 7707 . . . . . 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 734 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  =  suc  x )
2726fveq2d 5691 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
28 r1suc 7652 . . . . . . . . 9  |-  ( x  e.  On  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
2928ad2antrl 709 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
3027, 29eqtrd 2436 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e. WUni )
328adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  e.  On )
33 sucidg 4619 . . . . . . . . . . 11  |-  ( x  e.  On  ->  x  e.  suc  x )
3433ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  suc  x )
3534, 26eleqtrrd 2481 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  A )
36 r1ord 7662 . . . . . . . . 9  |-  ( A  e.  On  ->  (
x  e.  A  -> 
( R1 `  x
)  e.  ( R1
`  A ) ) )
3732, 35, 36sylc 58 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  x )  e.  ( R1 `  A ) )
3831, 37wunpw 8538 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ~P ( R1 `  x )  e.  ( R1 `  A
) )
3930, 38eqeltrd 2478 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e.  ( R1 `  A ) )
4039rexlimdvaa 2791 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  e.  ( R1
`  A ) ) )
4125, 40mtod 170 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  E. x  e.  On  A  =  suc  x )
42 ioran 477 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
4316, 41, 42sylanbrc 646 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) )
44 dflim3 4786 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
4510, 43, 44sylanbrc 646 . 2  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Lim  A )
46 r1limwun 8567 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni
)
4745, 46impbida 806 1  |-  ( A  e.  V  ->  (
( R1 `  A
)  e. WUni  <->  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   (/)c0 3588   ~Pcpw 3759   U.cuni 3975   Ord word 4540   Oncon0 4541   Lim wlim 4542   suc csuc 4543   dom cdm 4837   "cima 4840    Fn wfn 5408   ` cfv 5413   R1cr1 7644  WUnicwun 8531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647  df-wun 8533
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