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Theorem onuninsuci 4631
Description: A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
Hypothesis
Ref Expression
onssi.1  |-  A  e.  On
Assertion
Ref Expression
onuninsuci  |-  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x )
Distinct variable group:    x, A

Proof of Theorem onuninsuci
StepHypRef Expression
1 onssi.1 . . . . . . 7  |-  A  e.  On
21onirri 4499 . . . . . 6  |-  -.  A  e.  A
3 id 19 . . . . . . . 8  |-  ( A  =  U. A  ->  A  =  U. A )
4 df-suc 4398 . . . . . . . . . . . 12  |-  suc  x  =  ( x  u. 
{ x } )
54eqeq2i 2293 . . . . . . . . . . 11  |-  ( A  =  suc  x  <->  A  =  ( x  u.  { x } ) )
6 unieq 3836 . . . . . . . . . . 11  |-  ( A  =  ( x  u. 
{ x } )  ->  U. A  =  U. ( x  u.  { x } ) )
75, 6sylbi 187 . . . . . . . . . 10  |-  ( A  =  suc  x  ->  U. A  =  U. ( x  u.  { x } ) )
8 uniun 3846 . . . . . . . . . . 11  |-  U. (
x  u.  { x } )  =  ( U. x  u.  U. { x } )
9 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
109unisn 3843 . . . . . . . . . . . 12  |-  U. {
x }  =  x
1110uneq2i 3326 . . . . . . . . . . 11  |-  ( U. x  u.  U. { x } )  =  ( U. x  u.  x
)
128, 11eqtri 2303 . . . . . . . . . 10  |-  U. (
x  u.  { x } )  =  ( U. x  u.  x
)
137, 12syl6eq 2331 . . . . . . . . 9  |-  ( A  =  suc  x  ->  U. A  =  ( U. x  u.  x
) )
14 tron 4415 . . . . . . . . . . . 12  |-  Tr  On
15 eleq1 2343 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
161, 15mpbii 202 . . . . . . . . . . . 12  |-  ( A  =  suc  x  ->  suc  x  e.  On )
17 trsuc 4476 . . . . . . . . . . . 12  |-  ( ( Tr  On  /\  suc  x  e.  On )  ->  x  e.  On )
1814, 16, 17sylancr 644 . . . . . . . . . . 11  |-  ( A  =  suc  x  ->  x  e.  On )
19 eloni 4402 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  Ord  x )
20 ordtr 4406 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
2119, 20syl 15 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  Tr  x )
22 df-tr 4114 . . . . . . . . . . . 12  |-  ( Tr  x  <->  U. x  C_  x
)
2321, 22sylib 188 . . . . . . . . . . 11  |-  ( x  e.  On  ->  U. x  C_  x )
2418, 23syl 15 . . . . . . . . . 10  |-  ( A  =  suc  x  ->  U. x  C_  x )
25 ssequn1 3345 . . . . . . . . . 10  |-  ( U. x  C_  x  <->  ( U. x  u.  x )  =  x )
2624, 25sylib 188 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( U. x  u.  x )  =  x )
2713, 26eqtrd 2315 . . . . . . . 8  |-  ( A  =  suc  x  ->  U. A  =  x
)
283, 27sylan9eqr 2337 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  =  x )
299sucid 4471 . . . . . . . . 9  |-  x  e. 
suc  x
30 eleq2 2344 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( x  e.  A  <->  x  e.  suc  x ) )
3129, 30mpbiri 224 . . . . . . . 8  |-  ( A  =  suc  x  ->  x  e.  A )
3231adantr 451 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  x  e.  A )
3328, 32eqeltrd 2357 . . . . . 6  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  e.  A )
342, 33mto 167 . . . . 5  |-  -.  ( A  =  suc  x  /\  A  =  U. A )
3534imnani 412 . . . 4  |-  ( A  =  suc  x  ->  -.  A  =  U. A )
3635rexlimivw 2663 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  -.  A  =  U. A )
37 onuni 4584 . . . . 5  |-  ( A  e.  On  ->  U. A  e.  On )
381, 37ax-mp 8 . . . 4  |-  U. A  e.  On
391onuniorsuci 4630 . . . . 5  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
4039ori 364 . . . 4  |-  ( -.  A  =  U. A  ->  A  =  suc  U. A )
41 suceq 4457 . . . . . 6  |-  ( x  =  U. A  ->  suc  x  =  suc  U. A )
4241eqeq2d 2294 . . . . 5  |-  ( x  =  U. A  -> 
( A  =  suc  x 
<->  A  =  suc  U. A ) )
4342rspcev 2884 . . . 4  |-  ( ( U. A  e.  On  /\  A  =  suc  U. A )  ->  E. x  e.  On  A  =  suc  x )
4438, 40, 43sylancr 644 . . 3  |-  ( -.  A  =  U. A  ->  E. x  e.  On  A  =  suc  x )
4536, 44impbii 180 . 2  |-  ( E. x  e.  On  A  =  suc  x  <->  -.  A  =  U. A )
4645con2bii 322 1  |-  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150    C_ wss 3152   {csn 3640   U.cuni 3827   Tr wtr 4113   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  orduninsuc  4634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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