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Theorem ordunisuc2 4635
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 4634 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
2 ralnex 2553 . . 3  |-  ( A. x  e.  On  -.  A  =  suc  x  <->  -.  E. x  e.  On  A  =  suc  x )
3 suceloni 4604 . . . . . . . . . 10  |-  ( x  e.  On  ->  suc  x  e.  On )
4 eloni 4402 . . . . . . . . . 10  |-  ( suc  x  e.  On  ->  Ord 
suc  x )
53, 4syl 15 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  suc  x )
6 ordtri3 4428 . . . . . . . . 9  |-  ( ( Ord  A  /\  Ord  suc  x )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
75, 6sylan2 460 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
87con2bid 319 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  -.  A  =  suc  x ) )
9 onnbtwn 4484 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  -.  ( x  e.  A  /\  A  e.  suc  x ) )
10 imnan 411 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  ->  -.  A  e.  suc  x )  <->  -.  (
x  e.  A  /\  A  e.  suc  x ) )
119, 10sylibr 203 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  e.  A  ->  -.  A  e.  suc  x ) )
1211con2d 107 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( A  e.  suc  x  ->  -.  x  e.  A
) )
13 pm2.21 100 . . . . . . . . . . 11  |-  ( -.  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) )
1412, 13syl6 29 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( A  e.  suc  x  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
1514adantl 452 . . . . . . . . 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 10 . . . . . . . . 9  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( suc  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
1815, 17jaod 369 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  ->  (
x  e.  A  ->  suc  x  e.  A ) ) )
19 eloni 4402 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  Ord  x )
20 ordtri2or 4488 . . . . . . . . . . . . . 14  |-  ( ( Ord  x  /\  Ord  A )  ->  ( x  e.  A  \/  A  C_  x ) )
2119, 20sylan 457 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  e.  A  \/  A  C_  x ) )
2221ancoms 439 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
x  e.  A  \/  A  C_  x ) )
2322orcomd 377 . . . . . . . . . . 11  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  \/  x  e.  A ) )
2423adantr 451 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  \/  x  e.  A ) )
25 ordsssuc2 4481 . . . . . . . . . . . . 13  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  <->  A  e.  suc  x ) )
2625biimpd 198 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
2726adantr 451 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
28 simpr 447 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  (
x  e.  A  ->  suc  x  e.  A ) )
2927, 28orim12d 811 . . . . . . . . . 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 14 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  e.  suc  x  \/ 
suc  x  e.  A
) )
3130ex 423 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( x  e.  A  ->  suc  x  e.  A
)  ->  ( A  e.  suc  x  \/  suc  x  e.  A )
) )
3218, 31impbid 183 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
338, 32bitr3d 246 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( -.  A  =  suc  x 
<->  ( x  e.  A  ->  suc  x  e.  A
) ) )
3433pm5.74da 668 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) ) )
35 impexp 433 . . . . . 6  |-  ( ( ( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) )
36 simpr 447 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  e.  A )  ->  x  e.  A )
37 ordelon 4416 . . . . . . . . . 10  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
3837ex 423 . . . . . . . . 9  |-  ( Ord 
A  ->  ( x  e.  A  ->  x  e.  On ) )
3938ancrd 537 . . . . . . . 8  |-  ( Ord 
A  ->  ( x  e.  A  ->  ( x  e.  On  /\  x  e.  A ) ) )
4036, 39impbid2 195 . . . . . . 7  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  A )  <->  x  e.  A ) )
4140imbi1d 308 . . . . . 6  |-  ( Ord 
A  ->  ( (
( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4235, 41syl5bbr 250 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4334, 42bitrd 244 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4443ralbidv2 2565 . . 3  |-  ( Ord 
A  ->  ( A. x  e.  On  -.  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
452, 44syl5bbr 250 . 2  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
461, 45bitrd 244 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  dflim4  4639  limsuc2  27137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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