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Theorem onuninsuci 6450
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 4821 . . . . . 6  |-  -.  A  e.  A
3 id 22 . . . . . . . 8  |-  ( A  =  U. A  ->  A  =  U. A )
4 df-suc 4721 . . . . . . . . . . . 12  |-  suc  x  =  ( x  u. 
{ x } )
54eqeq2i 2451 . . . . . . . . . . 11  |-  ( A  =  suc  x  <->  A  =  ( x  u.  { x } ) )
6 unieq 4096 . . . . . . . . . . 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 4107 . . . . . . . . . . 11  |-  U. (
x  u.  { x } )  =  ( U. x  u.  U. { x } )
9 vex 2973 . . . . . . . . . . . . 13  |-  x  e. 
_V
109unisn 4103 . . . . . . . . . . . 12  |-  U. {
x }  =  x
1110uneq2i 3504 . . . . . . . . . . 11  |-  ( U. x  u.  U. { x } )  =  ( U. x  u.  x
)
128, 11eqtri 2461 . . . . . . . . . 10  |-  U. (
x  u.  { x } )  =  ( U. x  u.  x
)
137, 12syl6eq 2489 . . . . . . . . 9  |-  ( A  =  suc  x  ->  U. A  =  ( U. x  u.  x
) )
14 tron 4738 . . . . . . . . . . . 12  |-  Tr  On
15 eleq1 2501 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
161, 15mpbii 211 . . . . . . . . . . . 12  |-  ( A  =  suc  x  ->  suc  x  e.  On )
17 trsuc 4799 . . . . . . . . . . . 12  |-  ( ( Tr  On  /\  suc  x  e.  On )  ->  x  e.  On )
1814, 16, 17sylancr 658 . . . . . . . . . . 11  |-  ( A  =  suc  x  ->  x  e.  On )
19 eloni 4725 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  Ord  x )
20 ordtr 4729 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
2119, 20syl 16 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  Tr  x )
22 df-tr 4383 . . . . . . . . . . . 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 3523 . . . . . . . . . 10  |-  ( U. x  C_  x  <->  ( U. x  u.  x )  =  x )
2624, 25sylib 196 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( U. x  u.  x )  =  x )
2713, 26eqtrd 2473 . . . . . . . 8  |-  ( A  =  suc  x  ->  U. A  =  x
)
283, 27sylan9eqr 2495 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  =  x )
299sucid 4794 . . . . . . . . 9  |-  x  e. 
suc  x
30 eleq2 2502 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( x  e.  A  <->  x  e.  suc  x ) )
3129, 30mpbiri 233 . . . . . . . 8  |-  ( A  =  suc  x  ->  x  e.  A )
3231adantr 462 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  x  e.  A )
3328, 32eqeltrd 2515 . . . . . 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 2835 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  -.  A  =  U. A )
37 onuni 6403 . . . . 5  |-  ( A  e.  On  ->  U. A  e.  On )
381, 37ax-mp 5 . . . 4  |-  U. A  e.  On
391onuniorsuci 6449 . . . . 5  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
4039ori 375 . . . 4  |-  ( -.  A  =  U. A  ->  A  =  suc  U. A )
41 suceq 4780 . . . . . 6  |-  ( x  =  U. A  ->  suc  x  =  suc  U. A )
4241eqeq2d 2452 . . . . 5  |-  ( x  =  U. A  -> 
( A  =  suc  x 
<->  A  =  suc  U. A ) )
4342rspcev 3070 . . . 4  |-  ( ( U. A  e.  On  /\  A  =  suc  U. A )  ->  E. x  e.  On  A  =  suc  x )
4438, 40, 43sylancr 658 . . 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 1364    e. wcel 1761   E.wrex 2714    u. cun 3323    C_ wss 3325   {csn 3874   U.cuni 4088   Tr wtr 4382   Ord word 4714   Oncon0 4715   suc csuc 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721
This theorem is referenced by:  orduninsuc  6453
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