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Theorem r1wunlim 8446
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 447 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e. WUni
)
21wun0 8427 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  (/)  e.  ( R1 `  A ) )
3 elfvdm 5634 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  A  e.  dom  R1 )
42, 3syl 15 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  dom  R1 )
5 r1fnon 7526 . . . . . 6  |-  R1  Fn  On
6 fndm 5422 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
75, 6ax-mp 8 . . . . 5  |-  dom  R1  =  On
84, 7syl6eleq 2448 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  On )
9 eloni 4481 . . . 4  |-  ( A  e.  On  ->  Ord  A )
108, 9syl 15 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Ord  A )
11 n0i 3536 . . . . . 6  |-  ( (/)  e.  ( R1 `  A
)  ->  -.  ( R1 `  A )  =  (/) )
122, 11syl 15 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  =  (/) )
13 fveq2 5605 . . . . . 6  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
14 r10 7527 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1513, 14syl6eq 2406 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
1612, 15nsyl 113 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  A  =  (/) )
17 suceloni 4683 . . . . . . . 8  |-  ( A  e.  On  ->  suc  A  e.  On )
188, 17syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  suc  A  e.  On )
19 sucidg 4549 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  suc  A )
208, 19syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  A  e.  suc  A )
21 r1ord 7539 . . . . . . 7  |-  ( suc 
A  e.  On  ->  ( A  e.  suc  A  ->  ( R1 `  A
)  e.  ( R1
`  suc  A )
) )
2218, 20, 21sylc 56 . . . . . 6  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
23 r1elwf 7555 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
24 wfelirr 7584 . . . . . 6  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  ->  -.  ( R1 `  A
)  e.  ( R1
`  A ) )
2522, 23, 243syl 18 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( R1 `  A )  e.  ( R1 `  A ) )
26 simprr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  =  suc  x )
2726fveq2d 5609 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
28 r1suc 7529 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
2928ad2antrl 708 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
3027, 29eqtrd 2390 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e. WUni )
328adantr 451 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  A  e.  On )
33 sucidg 4549 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  x  e.  suc  x )
3433ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  suc  x )
3534, 26eleqtrrd 2435 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  x  e.  A )
36 r1ord 7539 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
x  e.  A  -> 
( R1 `  x
)  e.  ( R1
`  A ) ) )
3732, 35, 36sylc 56 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  x )  e.  ( R1 `  A ) )
3831, 37wunpw 8416 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ~P ( R1 `  x )  e.  ( R1 `  A
) )
3930, 38eqeltrd 2432 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  (
x  e.  On  /\  A  =  suc  x ) )  ->  ( R1 `  A )  e.  ( R1 `  A ) )
4039expr 598 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( R1 `  A
)  e. WUni )  /\  x  e.  On )  ->  ( A  =  suc  x  -> 
( R1 `  A
)  e.  ( R1
`  A ) ) )
4140rexlimdva 2743 . . . . 5  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  e.  ( R1
`  A ) ) )
4225, 41mtod 168 . . . 4  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  E. x  e.  On  A  =  suc  x )
43 ioran 476 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
4416, 42, 43sylanbrc 645 . . 3  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) )
45 dflim3 4717 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
4610, 44, 45sylanbrc 645 . 2  |-  ( ( A  e.  V  /\  ( R1 `  A )  e. WUni )  ->  Lim  A )
47 r1limwun 8445 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni
)
4846, 47impbida 805 1  |-  ( A  e.  V  ->  (
( R1 `  A
)  e. WUni  <->  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   (/)c0 3531   ~Pcpw 3701   U.cuni 3906   Ord word 4470   Oncon0 4471   Lim wlim 4472   suc csuc 4473   dom cdm 4768   "cima 4771    Fn wfn 5329   ` cfv 5334   R1cr1 7521  WUnicwun 8409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-reg 7393  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-recs 6472  df-rdg 6507  df-r1 7523  df-rank 7524  df-wun 8411
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