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Theorem onuninsuci 6574
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 4898 . . . . . 6  |-  -.  A  e.  A
3 id 22 . . . . . . . 8  |-  ( A  =  U. A  ->  A  =  U. A )
4 df-suc 4798 . . . . . . . . . . . 12  |-  suc  x  =  ( x  u. 
{ x } )
54eqeq2i 2400 . . . . . . . . . . 11  |-  ( A  =  suc  x  <->  A  =  ( x  u.  { x } ) )
6 unieq 4171 . . . . . . . . . . 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 4182 . . . . . . . . . . 11  |-  U. (
x  u.  { x } )  =  ( U. x  u.  U. { x } )
9 vex 3037 . . . . . . . . . . . . 13  |-  x  e. 
_V
109unisn 4178 . . . . . . . . . . . 12  |-  U. {
x }  =  x
1110uneq2i 3569 . . . . . . . . . . 11  |-  ( U. x  u.  U. { x } )  =  ( U. x  u.  x
)
128, 11eqtri 2411 . . . . . . . . . 10  |-  U. (
x  u.  { x } )  =  ( U. x  u.  x
)
137, 12syl6eq 2439 . . . . . . . . 9  |-  ( A  =  suc  x  ->  U. A  =  ( U. x  u.  x
) )
14 tron 4815 . . . . . . . . . . . 12  |-  Tr  On
15 eleq1 2454 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
161, 15mpbii 211 . . . . . . . . . . . 12  |-  ( A  =  suc  x  ->  suc  x  e.  On )
17 trsuc 4876 . . . . . . . . . . . 12  |-  ( ( Tr  On  /\  suc  x  e.  On )  ->  x  e.  On )
1814, 16, 17sylancr 661 . . . . . . . . . . 11  |-  ( A  =  suc  x  ->  x  e.  On )
19 eloni 4802 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  Ord  x )
20 ordtr 4806 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
2119, 20syl 16 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  Tr  x )
22 df-tr 4461 . . . . . . . . . . . 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 3588 . . . . . . . . . 10  |-  ( U. x  C_  x  <->  ( U. x  u.  x )  =  x )
2624, 25sylib 196 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( U. x  u.  x )  =  x )
2713, 26eqtrd 2423 . . . . . . . 8  |-  ( A  =  suc  x  ->  U. A  =  x
)
283, 27sylan9eqr 2445 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  =  x )
299sucid 4871 . . . . . . . . 9  |-  x  e. 
suc  x
30 eleq2 2455 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( x  e.  A  <->  x  e.  suc  x ) )
3129, 30mpbiri 233 . . . . . . . 8  |-  ( A  =  suc  x  ->  x  e.  A )
3231adantr 463 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  x  e.  A )
3328, 32eqeltrd 2470 . . . . . 6  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  e.  A )
342, 33mto 176 . . . . 5  |-  -.  ( A  =  suc  x  /\  A  =  U. A )
3534imnani 421 . . . 4  |-  ( A  =  suc  x  ->  -.  A  =  U. A )
3635rexlimivw 2871 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  -.  A  =  U. A )
37 onuni 6527 . . . . 5  |-  ( A  e.  On  ->  U. A  e.  On )
381, 37ax-mp 5 . . . 4  |-  U. A  e.  On
391onuniorsuci 6573 . . . . 5  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
4039ori 373 . . . 4  |-  ( -.  A  =  U. A  ->  A  =  suc  U. A )
41 suceq 4857 . . . . . 6  |-  ( x  =  U. A  ->  suc  x  =  suc  U. A )
4241eqeq2d 2396 . . . . 5  |-  ( x  =  U. A  -> 
( A  =  suc  x 
<->  A  =  suc  U. A ) )
4342rspcev 3135 . . . 4  |-  ( ( U. A  e.  On  /\  A  =  suc  U. A )  ->  E. x  e.  On  A  =  suc  x )
4438, 40, 43sylancr 661 . . 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 330 1  |-  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733    u. cun 3387    C_ wss 3389   {csn 3944   U.cuni 4163   Tr wtr 4460   Ord word 4791   Oncon0 4792   suc csuc 4794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-suc 4798
This theorem is referenced by:  orduninsuc  6577
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