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Theorem onuninsuci 6659
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 4984 . . . . . 6  |-  -.  A  e.  A
3 id 22 . . . . . . . 8  |-  ( A  =  U. A  ->  A  =  U. A )
4 df-suc 4884 . . . . . . . . . . . 12  |-  suc  x  =  ( x  u. 
{ x } )
54eqeq2i 2485 . . . . . . . . . . 11  |-  ( A  =  suc  x  <->  A  =  ( x  u.  { x } ) )
6 unieq 4253 . . . . . . . . . . 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 4264 . . . . . . . . . . 11  |-  U. (
x  u.  { x } )  =  ( U. x  u.  U. { x } )
9 vex 3116 . . . . . . . . . . . . 13  |-  x  e. 
_V
109unisn 4260 . . . . . . . . . . . 12  |-  U. {
x }  =  x
1110uneq2i 3655 . . . . . . . . . . 11  |-  ( U. x  u.  U. { x } )  =  ( U. x  u.  x
)
128, 11eqtri 2496 . . . . . . . . . 10  |-  U. (
x  u.  { x } )  =  ( U. x  u.  x
)
137, 12syl6eq 2524 . . . . . . . . 9  |-  ( A  =  suc  x  ->  U. A  =  ( U. x  u.  x
) )
14 tron 4901 . . . . . . . . . . . 12  |-  Tr  On
15 eleq1 2539 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
161, 15mpbii 211 . . . . . . . . . . . 12  |-  ( A  =  suc  x  ->  suc  x  e.  On )
17 trsuc 4962 . . . . . . . . . . . 12  |-  ( ( Tr  On  /\  suc  x  e.  On )  ->  x  e.  On )
1814, 16, 17sylancr 663 . . . . . . . . . . 11  |-  ( A  =  suc  x  ->  x  e.  On )
19 eloni 4888 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  Ord  x )
20 ordtr 4892 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
2119, 20syl 16 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  Tr  x )
22 df-tr 4541 . . . . . . . . . . . 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 3674 . . . . . . . . . 10  |-  ( U. x  C_  x  <->  ( U. x  u.  x )  =  x )
2624, 25sylib 196 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( U. x  u.  x )  =  x )
2713, 26eqtrd 2508 . . . . . . . 8  |-  ( A  =  suc  x  ->  U. A  =  x
)
283, 27sylan9eqr 2530 . . . . . . 7  |-  ( ( A  =  suc  x  /\  A  =  U. A )  ->  A  =  x )
299sucid 4957 . . . . . . . . 9  |-  x  e. 
suc  x
30 eleq2 2540 . . . . . . . . 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 2555 . . . . . 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 2952 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  -.  A  =  U. A )
37 onuni 6612 . . . . 5  |-  ( A  e.  On  ->  U. A  e.  On )
381, 37ax-mp 5 . . . 4  |-  U. A  e.  On
391onuniorsuci 6658 . . . . 5  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
4039ori 375 . . . 4  |-  ( -.  A  =  U. A  ->  A  =  suc  U. A )
41 suceq 4943 . . . . . 6  |-  ( x  =  U. A  ->  suc  x  =  suc  U. A )
4241eqeq2d 2481 . . . . 5  |-  ( x  =  U. A  -> 
( A  =  suc  x 
<->  A  =  suc  U. A ) )
4342rspcev 3214 . . . 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 1379    e. wcel 1767   E.wrex 2815    u. cun 3474    C_ wss 3476   {csn 4027   U.cuni 4245   Tr wtr 4540   Ord word 4877   Oncon0 4878   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884
This theorem is referenced by:  orduninsuc  6662
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