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Theorem ordunisuc2 4816
Description: An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
ordunisuc2  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
Distinct variable group:    x, A

Proof of Theorem ordunisuc2
StepHypRef Expression
1 orduninsuc 4815 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
2 ralnex 2707 . . 3  |-  ( A. x  e.  On  -.  A  =  suc  x  <->  -.  E. x  e.  On  A  =  suc  x )
3 suceloni 4785 . . . . . . . . . 10  |-  ( x  e.  On  ->  suc  x  e.  On )
4 eloni 4583 . . . . . . . . . 10  |-  ( suc  x  e.  On  ->  Ord 
suc  x )
53, 4syl 16 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  suc  x )
6 ordtri3 4609 . . . . . . . . 9  |-  ( ( Ord  A  /\  Ord  suc  x )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
75, 6sylan2 461 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
87con2bid 320 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  -.  A  =  suc  x ) )
9 onnbtwn 4665 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  -.  ( x  e.  A  /\  A  e.  suc  x ) )
10 imnan 412 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  ->  -.  A  e.  suc  x )  <->  -.  (
x  e.  A  /\  A  e.  suc  x ) )
119, 10sylibr 204 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  e.  A  ->  -.  A  e.  suc  x ) )
1211con2d 109 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( A  e.  suc  x  ->  -.  x  e.  A
) )
13 pm2.21 102 . . . . . . . . . . 11  |-  ( -.  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) )
1412, 13syl6 31 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( A  e.  suc  x  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
1514adantl 453 . . . . . . . . 9  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  e.  suc  x  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
16 ax-1 5 . . . . . . . . . 10  |-  ( suc  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) )
1716a1i 11 . . . . . . . . 9  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( suc  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
1815, 17jaod 370 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  ->  (
x  e.  A  ->  suc  x  e.  A ) ) )
19 eloni 4583 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  Ord  x )
20 ordtri2or 4669 . . . . . . . . . . . . . 14  |-  ( ( Ord  x  /\  Ord  A )  ->  ( x  e.  A  \/  A  C_  x ) )
2119, 20sylan 458 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  e.  A  \/  A  C_  x ) )
2221ancoms 440 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
x  e.  A  \/  A  C_  x ) )
2322orcomd 378 . . . . . . . . . . 11  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  \/  x  e.  A ) )
2423adantr 452 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  \/  x  e.  A ) )
25 ordsssuc2 4662 . . . . . . . . . . . . 13  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  <->  A  e.  suc  x ) )
2625biimpd 199 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
2726adantr 452 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
28 simpr 448 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  (
x  e.  A  ->  suc  x  e.  A ) )
2927, 28orim12d 812 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  (
( A  C_  x  \/  x  e.  A
)  ->  ( A  e.  suc  x  \/  suc  x  e.  A )
) )
3024, 29mpd 15 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  e.  suc  x  \/ 
suc  x  e.  A
) )
3130ex 424 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( x  e.  A  ->  suc  x  e.  A
)  ->  ( A  e.  suc  x  \/  suc  x  e.  A )
) )
3218, 31impbid 184 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
338, 32bitr3d 247 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( -.  A  =  suc  x 
<->  ( x  e.  A  ->  suc  x  e.  A
) ) )
3433pm5.74da 669 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) ) )
35 impexp 434 . . . . . 6  |-  ( ( ( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) )
36 simpr 448 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  e.  A )  ->  x  e.  A )
37 ordelon 4597 . . . . . . . . . 10  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
3837ex 424 . . . . . . . . 9  |-  ( Ord 
A  ->  ( x  e.  A  ->  x  e.  On ) )
3938ancrd 538 . . . . . . . 8  |-  ( Ord 
A  ->  ( x  e.  A  ->  ( x  e.  On  /\  x  e.  A ) ) )
4036, 39impbid2 196 . . . . . . 7  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  A )  <->  x  e.  A ) )
4140imbi1d 309 . . . . . 6  |-  ( Ord 
A  ->  ( (
( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4235, 41syl5bbr 251 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4334, 42bitrd 245 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4443ralbidv2 2719 . . 3  |-  ( Ord 
A  ->  ( A. x  e.  On  -.  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
452, 44syl5bbr 251 . 2  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
461, 45bitrd 245 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  dflim4  4820  limsuc2  27096
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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