MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orduninsuc Structured version   Unicode version

Theorem orduninsuc 6557
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 6503 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 id 22 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  A  =  if ( A  e.  On ,  A ,  (/) ) )
3 unieq 4200 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  U. A  =  U. if ( A  e.  On ,  A ,  (/) ) )
42, 3eqeq12d 2473 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  U. A  <->  if ( A  e.  On ,  A ,  (/) )  = 
U. if ( A  e.  On ,  A ,  (/) ) ) )
5 eqeq1 2455 . . . . . . 7  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  =  suc  x  <->  if ( A  e.  On ,  A ,  (/) )  =  suc  x ) )
65rexbidv 2855 . . . . . 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 294 . . . . 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 321 . . . 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 4873 . . . . . 6  |-  (/)  e.  On
109elimel 3953 . . . . 5  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1110onuninsuci 6554 . . . 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 3942 . . 3  |-  ( A  e.  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
13 unon 6545 . . . . . 6  |-  U. On  =  On
1413eqcomi 2464 . . . . 5  |-  On  =  U. On
15 onprc 6499 . . . . . . . 8  |-  -.  On  e.  _V
16 vex 3074 . . . . . . . . . 10  |-  x  e. 
_V
1716sucex 6525 . . . . . . . . 9  |-  suc  x  e.  _V
18 eleq1 2523 . . . . . . . . 9  |-  ( On  =  suc  x  -> 
( On  e.  _V  <->  suc  x  e.  _V )
)
1917, 18mpbiri 233 . . . . . . . 8  |-  ( On  =  suc  x  ->  On  e.  _V )
2015, 19mto 176 . . . . . . 7  |-  -.  On  =  suc  x
2120a1i 11 . . . . . 6  |-  ( x  e.  On  ->  -.  On  =  suc  x )
2221nrex 2917 . . . . 5  |-  -.  E. x  e.  On  On  =  suc  x
2314, 222th 239 . . . 4  |-  ( On  =  U. On  <->  -.  E. x  e.  On  On  =  suc  x )
24 id 22 . . . . . 6  |-  ( A  =  On  ->  A  =  On )
25 unieq 4200 . . . . . 6  |-  ( A  =  On  ->  U. A  =  U. On )
2624, 25eqeq12d 2473 . . . . 5  |-  ( A  =  On  ->  ( A  =  U. A  <->  On  =  U. On ) )
27 eqeq1 2455 . . . . . . 7  |-  ( A  =  On  ->  ( A  =  suc  x  <->  On  =  suc  x ) )
2827rexbidv 2855 . . . . . 6  |-  ( A  =  On  ->  ( E. x  e.  On  A  =  suc  x  <->  E. x  e.  On  On  =  suc  x ) )
2928notbid 294 . . . . 5  |-  ( A  =  On  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  -.  E. x  e.  On  On  =  suc  x ) )
3026, 29bibi12d 321 . . . 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 233 . . 3  |-  ( A  =  On  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
3212, 31jaoi 379 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  =  U. A 
<->  -.  E. x  e.  On  A  =  suc  x ) )
331, 32sylbi 195 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3071   (/)c0 3738   ifcif 3892   U.cuni 4192   Ord word 4819   Oncon0 4820   suc csuc 4822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826
This theorem is referenced by:  ordunisuc2  6558  ordzsl  6559  dflim3  6561  nnsuc  6596
  Copyright terms: Public domain W3C validator