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Theorem ordunisuc2 6685
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 6684 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
2 ralnex 2868 . . 3  |-  ( A. x  e.  On  -.  A  =  suc  x  <->  -.  E. x  e.  On  A  =  suc  x )
3 suceloni 6654 . . . . . . . . . 10  |-  ( x  e.  On  ->  suc  x  e.  On )
4 eloni 5452 . . . . . . . . . 10  |-  ( suc  x  e.  On  ->  Ord 
suc  x )
53, 4syl 17 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  suc  x )
6 ordtri3 5478 . . . . . . . . 9  |-  ( ( Ord  A  /\  Ord  suc  x )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
75, 6sylan2 476 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
87con2bid 330 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  -.  A  =  suc  x ) )
9 onnbtwn 5533 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  -.  ( x  e.  A  /\  A  e.  suc  x ) )
10 imnan 423 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  ->  -.  A  e.  suc  x )  <->  -.  (
x  e.  A  /\  A  e.  suc  x ) )
119, 10sylibr 215 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  e.  A  ->  -.  A  e.  suc  x ) )
1211con2d 118 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( A  e.  suc  x  ->  -.  x  e.  A
) )
13 pm2.21 111 . . . . . . . . . . 11  |-  ( -.  x  e.  A  -> 
( x  e.  A  ->  suc  x  e.  A
) )
1412, 13syl6 34 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( A  e.  suc  x  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
1514adantl 467 . . . . . . . . 9  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  e.  suc  x  -> 
( x  e.  A  ->  suc  x  e.  A
) ) )
16 ax-1 6 . . . . . . . . . 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 381 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  ->  (
x  e.  A  ->  suc  x  e.  A ) ) )
19 eloni 5452 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  Ord  x )
20 ordtri2or 5537 . . . . . . . . . . . . . 14  |-  ( ( Ord  x  /\  Ord  A )  ->  ( x  e.  A  \/  A  C_  x ) )
2119, 20sylan 473 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  e.  A  \/  A  C_  x ) )
2221ancoms 454 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
x  e.  A  \/  A  C_  x ) )
2322orcomd 389 . . . . . . . . . . 11  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  \/  x  e.  A ) )
2423adantr 466 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  \/  x  e.  A ) )
25 ordsssuc2 5530 . . . . . . . . . . . . 13  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  <->  A  e.  suc  x ) )
2625biimpd 210 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
2726adantr 466 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
28 simpr 462 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  (
x  e.  A  ->  suc  x  e.  A ) )
2927, 28orim12d 846 . . . . . . . . . 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 435 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( x  e.  A  ->  suc  x  e.  A
)  ->  ( A  e.  suc  x  \/  suc  x  e.  A )
) )
3218, 31impbid 193 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
338, 32bitr3d 258 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( -.  A  =  suc  x 
<->  ( x  e.  A  ->  suc  x  e.  A
) ) )
3433pm5.74da 691 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) ) )
35 impexp 447 . . . . . 6  |-  ( ( ( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) )
36 simpr 462 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  e.  A )  ->  x  e.  A )
37 ordelon 5466 . . . . . . . . . 10  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
3837ex 435 . . . . . . . . 9  |-  ( Ord 
A  ->  ( x  e.  A  ->  x  e.  On ) )
3938ancrd 556 . . . . . . . 8  |-  ( Ord 
A  ->  ( x  e.  A  ->  ( x  e.  On  /\  x  e.  A ) ) )
4036, 39impbid2 207 . . . . . . 7  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  A )  <->  x  e.  A ) )
4140imbi1d 318 . . . . . 6  |-  ( Ord 
A  ->  ( (
( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4235, 41syl5bbr 262 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4334, 42bitrd 256 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4443ralbidv2 2857 . . 3  |-  ( Ord 
A  ->  ( A. x  e.  On  -.  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
452, 44syl5bbr 262 . 2  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
461, 45bitrd 256 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772    C_ wss 3436   U.cuni 4219   Ord word 5441   Oncon0 5442   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-suc 5448
This theorem is referenced by:  dflim4  6689  limsuc2  35869
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