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Theorem orduninsuc 4634
Description: An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
orduninsuc  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
Distinct variable group:    x, A

Proof of Theorem orduninsuc
StepHypRef Expression
1 ordeleqon 4580 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 id 19 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  A  =  if ( A  e.  On ,  A ,  (/) ) )
3 unieq 3836 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  U. A  =  U. if ( A  e.  On ,  A ,  (/) ) )
42, 3eqeq12d 2297 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  U. A  <->  if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) ) ) )
5 eqeq1 2289 . . . . . . 7  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  suc  x  <->  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
65rexbidv 2564 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( E. x  e.  On  A  =  suc  x  <->  E. x  e.  On  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
76notbid 285 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  -.  E. x  e.  On  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
84, 7bibi12d 312 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  =  U. A 
<->  -.  E. x  e.  On  A  =  suc  x )  <->  ( if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) )  <->  -.  E. x  e.  On  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) ) )
9 0elon 4445 . . . . . 6  |-  (/)  e.  On
109elimel 3617 . . . . 5  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1110onuninsuci 4631 . . . 4  |-  ( if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) )  <->  -.  E. x  e.  On  if ( A  e.  On ,  A ,  (/) )  =  suc  x )
128, 11dedth 3606 . . 3  |-  ( A  e.  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
13 unon 4622 . . . . . 6  |-  U. On  =  On
1413eqcomi 2287 . . . . 5  |-  On  =  U. On
15 onprc 4576 . . . . . . . 8  |-  -.  On  e.  _V
16 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
1716sucex 4602 . . . . . . . . 9  |-  suc  x  e.  _V
18 eleq1 2343 . . . . . . . . 9  |-  ( On  =  suc  x  -> 
( On  e.  _V  <->  suc  x  e.  _V )
)
1917, 18mpbiri 224 . . . . . . . 8  |-  ( On  =  suc  x  ->  On  e.  _V )
2015, 19mto 167 . . . . . . 7  |-  -.  On  =  suc  x
2120a1i 10 . . . . . 6  |-  ( x  e.  On  ->  -.  On  =  suc  x )
2221nrex 2645 . . . . 5  |-  -.  E. x  e.  On  On  =  suc  x
2314, 222th 230 . . . 4  |-  ( On  =  U. On  <->  -.  E. x  e.  On  On  =  suc  x )
24 id 19 . . . . . 6  |-  ( A  =  On  ->  A  =  On )
25 unieq 3836 . . . . . 6  |-  ( A  =  On  ->  U. A  =  U. On )
2624, 25eqeq12d 2297 . . . . 5  |-  ( A  =  On  ->  ( A  =  U. A  <->  On  =  U. On ) )
27 eqeq1 2289 . . . . . . 7  |-  ( A  =  On  ->  ( A  =  suc  x  <->  On  =  suc  x ) )
2827rexbidv 2564 . . . . . 6  |-  ( A  =  On  ->  ( E. x  e.  On  A  =  suc  x  <->  E. x  e.  On  On  =  suc  x ) )
2928notbid 285 . . . . 5  |-  ( A  =  On  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  -.  E. x  e.  On  On  =  suc  x ) )
3026, 29bibi12d 312 . . . 4  |-  ( A  =  On  ->  (
( A  =  U. A 
<->  -.  E. x  e.  On  A  =  suc  x )  <->  ( On  =  U. On  <->  -.  E. x  e.  On  On  =  suc  x ) ) )
3123, 30mpbiri 224 . . 3  |-  ( A  =  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
3212, 31jaoi 368 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  =  U. A 
<->  -.  E. x  e.  On  A  =  suc  x ) )
331, 32sylbi 187 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ifcif 3565   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordunisuc2  4635  ordzsl  4636  dflim3  4638  nnsuc  4673
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-pw 3627  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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