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Theorem onsucuni2 4641
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2356 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 472 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 eloni 4418 . . . . 5  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
4 ordsuc 4621 . . . . . . . 8  |-  ( Ord 
B  <->  Ord  suc  B )
5 ordunisuc 4639 . . . . . . . 8  |-  ( Ord 
B  ->  U. suc  B  =  B )
64, 5sylbir 204 . . . . . . 7  |-  ( Ord 
suc  B  ->  U. suc  B  =  B )
7 suceq 4473 . . . . . . 7  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
86, 7syl 15 . . . . . 6  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  suc  B
)
9 ordunisuc 4639 . . . . . 6  |-  ( Ord 
suc  B  ->  U. suc  suc 
B  =  suc  B
)
108, 9eqtr4d 2331 . . . . 5  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  U. suc  suc 
B )
112, 3, 103syl 18 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
12 unieq 3852 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
13 suceq 4473 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
1412, 13syl 15 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
15 suceq 4473 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
1615unieqd 3854 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
1714, 16eqeq12d 2310 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
1811, 17syl5ibr 212 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
1918anabsi7 792 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
20 eloni 4418 . . . 4  |-  ( A  e.  On  ->  Ord  A )
21 ordunisuc 4639 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
2220, 21syl 15 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
2322adantr 451 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
2419, 23eqtrd 2328 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  rankxplim3  7567  rankxpsuc  7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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