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Theorem ordunisuc2 6615
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 6614 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
2 ralnex 2847 . . 3  |-  ( A. x  e.  On  -.  A  =  suc  x  <->  -.  E. x  e.  On  A  =  suc  x )
3 suceloni 6584 . . . . . . . . . 10  |-  ( x  e.  On  ->  suc  x  e.  On )
4 eloni 4829 . . . . . . . . . 10  |-  ( suc  x  e.  On  ->  Ord 
suc  x )
53, 4syl 17 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  suc  x )
6 ordtri3 4855 . . . . . . . . 9  |-  ( ( Ord  A  /\  Ord  suc  x )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
75, 6sylan2 472 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  =  suc  x  <->  -.  ( A  e.  suc  x  \/ 
suc  x  e.  A
) ) )
87con2bid 327 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  -.  A  =  suc  x ) )
9 onnbtwn 4910 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  -.  ( x  e.  A  /\  A  e.  suc  x ) )
10 imnan 420 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  ->  -.  A  e.  suc  x )  <->  -.  (
x  e.  A  /\  A  e.  suc  x ) )
119, 10sylibr 212 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  e.  A  ->  -.  A  e.  suc  x ) )
1211con2d 115 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( A  e.  suc  x  ->  -.  x  e.  A
) )
13 pm2.21 108 . . . . . . . . . . 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 464 . . . . . . . . 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 378 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  ->  (
x  e.  A  ->  suc  x  e.  A ) ) )
19 eloni 4829 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  Ord  x )
20 ordtri2or 4914 . . . . . . . . . . . . . 14  |-  ( ( Ord  x  /\  Ord  A )  ->  ( x  e.  A  \/  A  C_  x ) )
2119, 20sylan 469 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  Ord  A )  ->  (
x  e.  A  \/  A  C_  x ) )
2221ancoms 451 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
x  e.  A  \/  A  C_  x ) )
2322orcomd 386 . . . . . . . . . . 11  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  \/  x  e.  A ) )
2423adantr 463 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  \/  x  e.  A ) )
25 ordsssuc2 4907 . . . . . . . . . . . . 13  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  <->  A  e.  suc  x ) )
2625biimpd 207 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
2726adantr 463 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  ( A  C_  x  ->  A  e.  suc  x ) )
28 simpr 459 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  On )  /\  ( x  e.  A  ->  suc  x  e.  A
) )  ->  (
x  e.  A  ->  suc  x  e.  A ) )
2927, 28orim12d 837 . . . . . . . . . 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 432 . . . . . . . 8  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( x  e.  A  ->  suc  x  e.  A
)  ->  ( A  e.  suc  x  \/  suc  x  e.  A )
) )
3218, 31impbid 191 . . . . . . 7  |-  ( ( Ord  A  /\  x  e.  On )  ->  (
( A  e.  suc  x  \/  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
338, 32bitr3d 255 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  On )  ->  ( -.  A  =  suc  x 
<->  ( x  e.  A  ->  suc  x  e.  A
) ) )
3433pm5.74da 685 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) ) )
35 impexp 444 . . . . . 6  |-  ( ( ( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) ) )
36 simpr 459 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  e.  A )  ->  x  e.  A )
37 ordelon 4843 . . . . . . . . . 10  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
3837ex 432 . . . . . . . . 9  |-  ( Ord 
A  ->  ( x  e.  A  ->  x  e.  On ) )
3938ancrd 552 . . . . . . . 8  |-  ( Ord 
A  ->  ( x  e.  A  ->  ( x  e.  On  /\  x  e.  A ) ) )
4036, 39impbid2 204 . . . . . . 7  |-  ( Ord 
A  ->  ( (
x  e.  On  /\  x  e.  A )  <->  x  e.  A ) )
4140imbi1d 315 . . . . . 6  |-  ( Ord 
A  ->  ( (
( x  e.  On  /\  x  e.  A )  ->  suc  x  e.  A )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4235, 41syl5bbr 259 . . . . 5  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  ( x  e.  A  ->  suc  x  e.  A ) )  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4334, 42bitrd 253 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  On  ->  -.  A  =  suc  x
)  <->  ( x  e.  A  ->  suc  x  e.  A ) ) )
4443ralbidv2 2836 . . 3  |-  ( Ord 
A  ->  ( A. x  e.  On  -.  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
452, 44syl5bbr 259 . 2  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  <->  A. x  e.  A  suc  x  e.  A ) )
461, 45bitrd 253 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 184    \/ wo 366    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752    C_ wss 3411   U.cuni 4188   Ord word 4818   Oncon0 4819   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825
This theorem is referenced by:  dflim4  6619  limsuc2  35312
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