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Theorem wunr1om 8594
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 7693 . . . . . 6  |-  R1  Fn  On
2 fnfun 5542 . . . . . 6  |-  ( R1  Fn  On  ->  Fun  R1 )
31, 2ax-mp 8 . . . . 5  |-  Fun  R1
4 fvelima 5778 . . . . 5  |-  ( ( Fun  R1  /\  y  e.  ( R1 " om ) )  ->  E. x  e.  om  ( R1 `  x )  =  y )
53, 4mpan 652 . . . 4  |-  ( y  e.  ( R1 " om )  ->  E. x  e.  om  ( R1 `  x )  =  y )
6 fveq2 5728 . . . . . . . 8  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
76eleq1d 2502 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( R1 `  x )  e.  U  <->  ( R1 `  (/) )  e.  U
) )
8 fveq2 5728 . . . . . . . 8  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
98eleq1d 2502 . . . . . . 7  |-  ( x  =  y  ->  (
( R1 `  x
)  e.  U  <->  ( R1 `  y )  e.  U
) )
10 fveq2 5728 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
1110eleq1d 2502 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( R1 `  x )  e.  U  <->  ( R1 `  suc  y
)  e.  U ) )
12 r10 7694 . . . . . . . 8  |-  ( R1
`  (/) )  =  (/)
13 wun0.1 . . . . . . . . 9  |-  ( ph  ->  U  e. WUni )
1413wun0 8593 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  U )
1512, 14syl5eqel 2520 . . . . . . 7  |-  ( ph  ->  ( R1 `  (/) )  e.  U )
1613adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  U  e. WUni )
17 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  ( R1 `  y )  e.  U
)
1816, 17wunpw 8582 . . . . . . . . 9  |-  ( (
ph  /\  ( R1 `  y )  e.  U
)  ->  ~P ( R1 `  y )  e.  U )
19 nnon 4851 . . . . . . . . . . 11  |-  ( y  e.  om  ->  y  e.  On )
20 r1suc 7696 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( y  e.  om  ->  ( R1 `  suc  y )  =  ~P ( R1
`  y ) )
2221eleq1d 2502 . . . . . . . . 9  |-  ( y  e.  om  ->  (
( R1 `  suc  y )  e.  U  <->  ~P ( R1 `  y
)  e.  U ) )
2318, 22syl5ibr 213 . . . . . . . 8  |-  ( y  e.  om  ->  (
( ph  /\  ( R1 `  y )  e.  U )  ->  ( R1 `  suc  y )  e.  U ) )
2423exp3a 426 . . . . . . 7  |-  ( y  e.  om  ->  ( ph  ->  ( ( R1
`  y )  e.  U  ->  ( R1 ` 
suc  y )  e.  U ) ) )
257, 9, 11, 15, 24finds2 4873 . . . . . 6  |-  ( x  e.  om  ->  ( ph  ->  ( R1 `  x )  e.  U
) )
26 eleq1 2496 . . . . . . 7  |-  ( ( R1 `  x )  =  y  ->  (
( R1 `  x
)  e.  U  <->  y  e.  U ) )
2726imbi2d 308 . . . . . 6  |-  ( ( R1 `  x )  =  y  ->  (
( ph  ->  ( R1
`  x )  e.  U )  <->  ( ph  ->  y  e.  U ) ) )
2825, 27syl5ibcom 212 . . . . 5  |-  ( x  e.  om  ->  (
( R1 `  x
)  =  y  -> 
( ph  ->  y  e.  U ) ) )
2928rexlimiv 2824 . . . 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 29 . 2  |-  ( ph  ->  ( y  e.  ( R1 " om )  ->  y  e.  U ) )
3231ssrdv 3354 1  |-  ( ph  ->  ( R1 " om )  C_  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   Oncon0 4581   suc csuc 4583   omcom 4845   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454   R1cr1 7688  WUnicwun 8575
This theorem is referenced by:  wunom  8595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-wun 8577
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