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Theorem orduninsuc 4713
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 4659 . 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 3915 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  U. A  =  U. if ( A  e.  On ,  A ,  (/) ) )
42, 3eqeq12d 2372 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  U. A  <->  if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) ) ) )
5 eqeq1 2364 . . . . . . 7  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  suc  x  <->  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
65rexbidv 2640 . . . . . 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 4524 . . . . . 6  |-  (/)  e.  On
109elimel 3693 . . . . 5  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1110onuninsuci 4710 . . . 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 3682 . . 3  |-  ( A  e.  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
13 unon 4701 . . . . . 6  |-  U. On  =  On
1413eqcomi 2362 . . . . 5  |-  On  =  U. On
15 onprc 4655 . . . . . . . 8  |-  -.  On  e.  _V
16 vex 2867 . . . . . . . . . 10  |-  x  e. 
_V
1716sucex 4681 . . . . . . . . 9  |-  suc  x  e.  _V
18 eleq1 2418 . . . . . . . . 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 2721 . . . . 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 3915 . . . . . 6  |-  ( A  =  On  ->  U. A  =  U. On )
2624, 25eqeq12d 2372 . . . . 5  |-  ( A  =  On  ->  ( A  =  U. A  <->  On  =  U. On ) )
27 eqeq1 2364 . . . . . . 7  |-  ( A  =  On  ->  ( A  =  suc  x  <->  On  =  suc  x ) )
2827rexbidv 2640 . . . . . 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 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   (/)c0 3531   ifcif 3641   U.cuni 3906   Ord word 4470   Oncon0 4471   suc csuc 4473
This theorem is referenced by:  ordunisuc2  4714  ordzsl  4715  dflim3  4717  nnsuc  4752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477
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