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Theorem orduninsuc 4790
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 4736 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 id 20 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  A  =  if ( A  e.  On ,  A ,  (/) ) )
3 unieq 3992 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  U. A  =  U. if ( A  e.  On ,  A ,  (/) ) )
42, 3eqeq12d 2426 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  U. A  <->  if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) ) ) )
5 eqeq1 2418 . . . . . . 7  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  suc  x  <->  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
65rexbidv 2695 . . . . . 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 286 . . . . 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 313 . . . 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 4602 . . . . . 6  |-  (/)  e.  On
109elimel 3759 . . . . 5  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1110onuninsuci 4787 . . . 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 3748 . . 3  |-  ( A  e.  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
13 unon 4778 . . . . . 6  |-  U. On  =  On
1413eqcomi 2416 . . . . 5  |-  On  =  U. On
15 onprc 4732 . . . . . . . 8  |-  -.  On  e.  _V
16 vex 2927 . . . . . . . . . 10  |-  x  e. 
_V
1716sucex 4758 . . . . . . . . 9  |-  suc  x  e.  _V
18 eleq1 2472 . . . . . . . . 9  |-  ( On  =  suc  x  -> 
( On  e.  _V  <->  suc  x  e.  _V )
)
1917, 18mpbiri 225 . . . . . . . 8  |-  ( On  =  suc  x  ->  On  e.  _V )
2015, 19mto 169 . . . . . . 7  |-  -.  On  =  suc  x
2120a1i 11 . . . . . 6  |-  ( x  e.  On  ->  -.  On  =  suc  x )
2221nrex 2776 . . . . 5  |-  -.  E. x  e.  On  On  =  suc  x
2314, 222th 231 . . . 4  |-  ( On  =  U. On  <->  -.  E. x  e.  On  On  =  suc  x )
24 id 20 . . . . . 6  |-  ( A  =  On  ->  A  =  On )
25 unieq 3992 . . . . . 6  |-  ( A  =  On  ->  U. A  =  U. On )
2624, 25eqeq12d 2426 . . . . 5  |-  ( A  =  On  ->  ( A  =  U. A  <->  On  =  U. On ) )
27 eqeq1 2418 . . . . . . 7  |-  ( A  =  On  ->  ( A  =  suc  x  <->  On  =  suc  x ) )
2827rexbidv 2695 . . . . . 6  |-  ( A  =  On  ->  ( E. x  e.  On  A  =  suc  x  <->  E. x  e.  On  On  =  suc  x ) )
2928notbid 286 . . . . 5  |-  ( A  =  On  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  -.  E. x  e.  On  On  =  suc  x ) )
3026, 29bibi12d 313 . . . 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 225 . . 3  |-  ( A  =  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
3212, 31jaoi 369 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  =  U. A 
<->  -.  E. x  e.  On  A  =  suc  x ) )
331, 32sylbi 188 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 177    \/ wo 358    = wceq 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924   (/)c0 3596   ifcif 3707   U.cuni 3983   Ord word 4548   Oncon0 4549   suc csuc 4551
This theorem is referenced by:  ordunisuc2  4791  ordzsl  4792  dflim3  4794  nnsuc  4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555
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