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Theorem wunr1om 9109
Description: A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypothesis
Ref Expression
wun0.1  |-  ( ph  ->  U  e. WUni )
Assertion
Ref Expression
wunr1om  |-  ( ph  ->  ( R1 " om )  C_  U )

Proof of Theorem wunr1om
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1fnon 8197 . . . . . 6  |-  R1  Fn  On
2 fnfun 5684 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 5 . . . . 5  |-  Fun  R1
4 fvelima 5926 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 670 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5872 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2536 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  U  <->  ( R1 `  (/) )  e.  U
) )
8 fveq2 5872 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2536 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  U  <->  ( R1 `  y )  e.  U
) )
10 fveq2 5872 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2536 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  U  <->  ( R1 `  suc  y
)  e.  U ) )
12 r10 8198 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 wun0.1 . . . . . . . . 9  |-  ( ph  ->  U  e. WUni )
1413wun0 9108 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  U )
1512, 14syl5eqel 2559 . . . . . . 7  |-  ( ph  ->  ( R1 `  (/) )  e.  U )
1613adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  U  e. WUni )
17 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  ( R1 `  y )  e.  U
)
1816, 17wunpw 9097 . . . . . . . . 9  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  ~P ( R1 `  y )  e.  U )
19 nnon 6701 . . . . . . . . . . 11  |-  ( y  e.  om  ->  y  e.  On )
20 r1suc 8200 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
2221eleq1d 2536 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  U  <->  ~P ( R1 `  y
)  e.  U ) )
2318, 22syl5ibr 221 . . . . . . . 8  |-  ( y  e.  om  ->  (
( ph  /\  ( R1 `  y )  e.  U )  ->  ( R1 `  suc  y )  e.  U ) )
2423expd 436 . . . . . . 7  |-  ( y  e.  om  ->  ( ph  ->  ( ( R1
`  y )  e.  U  ->  ( R1 ` 
suc  y )  e.  U ) ) )
257, 9, 11, 15, 24finds2 6723 . . . . . 6  |-  ( x  e.  om  ->  ( ph  ->  ( R1 `  x )  e.  U
) )
26 eleq1 2539 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  U  <->  y  e.  U ) )
2726imbi2d 316 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ph  ->  ( R1
`  x )  e.  U )  <->  ( ph  ->  y  e.  U ) ) )
2825, 27syl5ibcom 220 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ph  ->  y  e.  U ) ) )
2928rexlimiv 2953 . . . 4  |-  ( E. x  e.  om  ( R1 `  x )  =  y  ->  ( ph  ->  y  e.  U ) )
305, 29syl 16 . . 3  |-  ( y  e.  ( R1 " om )  ->  ( ph  ->  y  e.  U ) )
3130com12 31 . 2  |-  ( ph  ->  ( y  e.  ( R1 " om )  ->  y  e.  U ) )
3231ssrdv 3515 1  |-  ( ph  ->  ( R1 " om )  C_  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   Oncon0 4884   suc csuc 4886   "cima 5008   Fun wfun 5588    Fn wfn 5589   ` cfv 5594   omcom 6695   R1cr1 8192  WUnicwun 9090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-wun 9092
This theorem is referenced by:  wunom  9110
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