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Theorem onuninsuci 6451
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 4825 . . . . . 6  |-  -.  A  e.  A
3 id 22 . . . . . . . 8  |-  ( A  =  U. A  ->  A  =  U. A )
4 df-suc 4725 . . . . . . . . . . . 12  |-  suc  x  =  ( x  u. 
{ x } )
54eqeq2i 2453 . . . . . . . . . . 11  |-  ( A  =  suc  x  <->  A  =  ( x  u.  { x } ) )
6 unieq 4099 . . . . . . . . . . 11  |-  ( A  =  ( x  u. 
{ x } )  ->  U. A  =  U. ( x  u.  { x } ) )
75, 6sylbi 195 . . . . . . . . . 10  |-  ( A  =  suc  x  ->  U. A  =  U. ( x  u.  { x } ) )
8 uniun 4110 . . . . . . . . . . 11  |-  U. (
x  u.  { x } )  =  ( U. x  u.  U. { x } )
9 vex 2975 . . . . . . . . . . . . 13  |-  x  e. 
_V
109unisn 4106 . . . . . . . . . . . 12  |-  U. {
x }  =  x
1110uneq2i 3507 . . . . . . . . . . 11  |-  ( U. x  u.  U. { x } )  =  ( U. x  u.  x
)
128, 11eqtri 2463 . . . . . . . . . 10  |-  U. (
x  u.  { x } )  =  ( U. x  u.  x
)
137, 12syl6eq 2491 . . . . . . . . 9  |-  ( A  =  suc  x  ->  U. A  =  ( U. x  u.  x
) )
14 tron 4742 . . . . . . . . . . . 12  |-  Tr  On
15 eleq1 2503 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
161, 15mpbii 211 . . . . . . . . . . . 12  |-  ( A  =  suc  x  ->  suc  x  e.  On )
17 trsuc 4803 . . . . . . . . . . . 12  |-  ( ( Tr  On  /\  suc  x  e.  On )  ->  x  e.  On )
1814, 16, 17sylancr 663 . . . . . . . . . . 11  |-  ( A  =  suc  x  ->  x  e.  On )
19 eloni 4729 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  Ord  x )
20 ordtr 4733 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
2119, 20syl 16 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  Tr  x )
22 df-tr 4386 . . . . . . . . . . . 12  |-  ( Tr  x  <->  U. x  C_  x
)
2321, 22sylib 196 . . . . . . . . . . 11  |-  ( x  e.  On  ->  U. x  C_  x )
2418, 23syl 16 . . . . . . . . . 10  |-  ( A  =  suc  x  ->  U. x  C_  x )
25 ssequn1 3526 . . . . . . . . . 10  |-  ( U. x  C_  x  <->  ( U. x  u.  x )  =  x )
2624, 25sylib 196 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( U. x  u.  x )  =  x )
2713, 26eqtrd 2475 . . . . . . . 8  |-  ( A  =  suc  x  ->  U. A  =  x
)
283, 27sylan9eqr 2497 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  =  x )
299sucid 4798 . . . . . . . . 9  |-  x  e. 
suc  x
30 eleq2 2504 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( x  e.  A  <->  x  e.  suc  x ) )
3129, 30mpbiri 233 . . . . . . . 8  |-  ( A  =  suc  x  ->  x  e.  A )
3231adantr 465 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  x  e.  A )
3328, 32eqeltrd 2517 . . . . . 6  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  e.  A )
342, 33mto 176 . . . . 5  |-  -.  ( A  =  suc  x  /\  A  =  U. A )
3534imnani 423 . . . 4  |-  ( A  =  suc  x  ->  -.  A  =  U. A )
3635rexlimivw 2837 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  -.  A  =  U. A )
37 onuni 6404 . . . . 5  |-  ( A  e.  On  ->  U. A  e.  On )
381, 37ax-mp 5 . . . 4  |-  U. A  e.  On
391onuniorsuci 6450 . . . . 5  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
4039ori 375 . . . 4  |-  ( -.  A  =  U. A  ->  A  =  suc  U. A )
41 suceq 4784 . . . . . 6  |-  ( x  =  U. A  ->  suc  x  =  suc  U. A )
4241eqeq2d 2454 . . . . 5  |-  ( x  =  U. A  -> 
( A  =  suc  x 
<->  A  =  suc  U. A ) )
4342rspcev 3073 . . . 4  |-  ( ( U. A  e.  On  /\  A  =  suc  U. A )  ->  E. x  e.  On  A  =  suc  x )
4438, 40, 43sylancr 663 . . 3  |-  ( -.  A  =  U. A  ->  E. x  e.  On  A  =  suc  x )
4536, 44impbii 188 . 2  |-  ( E. x  e.  On  A  =  suc  x  <->  -.  A  =  U. A )
4645con2bii 332 1  |-  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    u. cun 3326    C_ wss 3328   {csn 3877   U.cuni 4091   Tr wtr 4385   Ord word 4718   Oncon0 4719   suc csuc 4721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725
This theorem is referenced by:  orduninsuc  6454
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