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Theorem onsucuni2 4548
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 2233 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 296 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 sucelon 4487 . . . . . . 7  |-  ( B  e.  On  <->  suc  B  e.  On )
4 eloni 4360 . . . . . . . . . 10  |-  ( B  e.  On  ->  Ord  B )
5 ordtr 4363 . . . . . . . . . 10  |-  ( Ord 
B  ->  Tr  B
)
64, 5syl 14 . . . . . . . . 9  |-  ( B  e.  On  ->  Tr  B )
7 unisucg 4399 . . . . . . . . 9  |-  ( B  e.  On  ->  ( Tr  B  <->  U. suc  B  =  B ) )
86, 7mpbid 146 . . . . . . . 8  |-  ( B  e.  On  ->  U. suc  B  =  B )
9 suceq 4387 . . . . . . . 8  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
108, 9syl 14 . . . . . . 7  |-  ( B  e.  On  ->  suc  U.
suc  B  =  suc  B )
113, 10sylbir 134 . . . . . 6  |-  ( suc 
B  e.  On  ->  suc  U. suc  B  =  suc  B )
12 eloni 4360 . . . . . . . 8  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
13 ordtr 4363 . . . . . . . 8  |-  ( Ord 
suc  B  ->  Tr  suc  B )
1412, 13syl 14 . . . . . . 7  |-  ( suc 
B  e.  On  ->  Tr 
suc  B )
15 unisucg 4399 . . . . . . 7  |-  ( suc 
B  e.  On  ->  ( Tr  suc  B  <->  U. suc  suc  B  =  suc  B ) )
1614, 15mpbid 146 . . . . . 6  |-  ( suc 
B  e.  On  ->  U.
suc  suc  B  =  suc  B )
1711, 16eqtr4d 2206 . . . . 5  |-  ( suc 
B  e.  On  ->  suc  U. suc  B  =  U. suc  suc  B )
182, 17syl 14 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
19 unieq 3805 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
20 suceq 4387 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
2119, 20syl 14 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
22 suceq 4387 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
2322unieqd 3807 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
2421, 23eqeq12d 2185 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
2518, 24syl5ibr 155 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
2625anabsi7 576 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
27 eloni 4360 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
28 ordtr 4363 . . . . 5  |-  ( Ord 
A  ->  Tr  A
)
2927, 28syl 14 . . . 4  |-  ( A  e.  On  ->  Tr  A )
30 unisucg 4399 . . . 4  |-  ( A  e.  On  ->  ( Tr  A  <->  U. suc  A  =  A ) )
3129, 30mpbid 146 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
3231adantr 274 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
3326, 32eqtrd 2203 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   U.cuni 3796   Tr wtr 4087   Ord word 4347   Oncon0 4348   suc csuc 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by:  nnsucpred  4601  nnpredcl  4607
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