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Theorem orduninsuc 4823
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 4769 . 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 4024 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  U. A  =  U. if ( A  e.  On ,  A ,  (/) ) )
42, 3eqeq12d 2450 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  U. A  <->  if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) ) ) )
5 eqeq1 2442 . . . . . . 7  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  suc  x  <->  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
65rexbidv 2726 . . . . . 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 4634 . . . . . 6  |-  (/)  e.  On
109elimel 3791 . . . . 5  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1110onuninsuci 4820 . . . 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 3780 . . 3  |-  ( A  e.  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
13 unon 4811 . . . . . 6  |-  U. On  =  On
1413eqcomi 2440 . . . . 5  |-  On  =  U. On
15 onprc 4765 . . . . . . . 8  |-  -.  On  e.  _V
16 vex 2959 . . . . . . . . . 10  |-  x  e. 
_V
1716sucex 4791 . . . . . . . . 9  |-  suc  x  e.  _V
18 eleq1 2496 . . . . . . . . 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 2808 . . . . 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 4024 . . . . . 6  |-  ( A  =  On  ->  U. A  =  U. On )
2624, 25eqeq12d 2450 . . . . 5  |-  ( A  =  On  ->  ( A  =  U. A  <->  On  =  U. On ) )
27 eqeq1 2442 . . . . . . 7  |-  ( A  =  On  ->  ( A  =  suc  x  <->  On  =  suc  x ) )
2827rexbidv 2726 . . . . . 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 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956   (/)c0 3628   ifcif 3739   U.cuni 4015   Ord word 4580   Oncon0 4581   suc csuc 4583
This theorem is referenced by:  ordunisuc2  4824  ordzsl  4825  dflim3  4827  nnsuc  4862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587
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