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Theorem onsucuni2 4814
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 2496 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 473 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 eloni 4591 . . . . 5  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
4 ordsuc 4794 . . . . . . . 8  |-  ( Ord 
B  <->  Ord  suc  B )
5 ordunisuc 4812 . . . . . . . 8  |-  ( Ord 
B  ->  U. suc  B  =  B )
64, 5sylbir 205 . . . . . . 7  |-  ( Ord 
suc  B  ->  U. suc  B  =  B )
7 suceq 4646 . . . . . . 7  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
86, 7syl 16 . . . . . 6  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  suc  B
)
9 ordunisuc 4812 . . . . . 6  |-  ( Ord 
suc  B  ->  U. suc  suc 
B  =  suc  B
)
108, 9eqtr4d 2471 . . . . 5  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  U. suc  suc 
B )
112, 3, 103syl 19 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
12 unieq 4024 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
13 suceq 4646 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
1412, 13syl 16 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
15 suceq 4646 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
1615unieqd 4026 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
1714, 16eqeq12d 2450 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
1811, 17syl5ibr 213 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
1918anabsi7 793 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
20 eloni 4591 . . . 4  |-  ( A  e.  On  ->  Ord  A )
21 ordunisuc 4812 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
2220, 21syl 16 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
2322adantr 452 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
2419, 23eqtrd 2468 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   U.cuni 4015   Ord word 4580   Oncon0 4581   suc csuc 4583
This theorem is referenced by:  rankxplim3  7805  rankxpsuc  7806
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587
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