ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onsucuni2 Unicode version

Theorem onsucuni2 4380
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 2150 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 292 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 sucelon 4320 . . . . . . 7  |-  ( B  e.  On  <->  suc  B  e.  On )
4 eloni 4202 . . . . . . . . . 10  |-  ( B  e.  On  ->  Ord  B )
5 ordtr 4205 . . . . . . . . . 10  |-  ( Ord 
B  ->  Tr  B
)
64, 5syl 14 . . . . . . . . 9  |-  ( B  e.  On  ->  Tr  B )
7 unisucg 4241 . . . . . . . . 9  |-  ( B  e.  On  ->  ( Tr  B  <->  U. suc  B  =  B ) )
86, 7mpbid 145 . . . . . . . 8  |-  ( B  e.  On  ->  U. suc  B  =  B )
9 suceq 4229 . . . . . . . 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 133 . . . . . 6  |-  ( suc 
B  e.  On  ->  suc  U. suc  B  =  suc  B )
12 eloni 4202 . . . . . . . 8  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
13 ordtr 4205 . . . . . . . 8  |-  ( Ord 
suc  B  ->  Tr  suc  B )
1412, 13syl 14 . . . . . . 7  |-  ( suc 
B  e.  On  ->  Tr 
suc  B )
15 unisucg 4241 . . . . . . 7  |-  ( suc 
B  e.  On  ->  ( Tr  suc  B  <->  U. suc  suc  B  =  suc  B ) )
1614, 15mpbid 145 . . . . . 6  |-  ( suc 
B  e.  On  ->  U.
suc  suc  B  =  suc  B )
1711, 16eqtr4d 2123 . . . . 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 3662 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
20 suceq 4229 . . . . . 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 4229 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
2322unieqd 3664 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
2421, 23eqeq12d 2102 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
2518, 24syl5ibr 154 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
2625anabsi7 548 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
27 eloni 4202 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
28 ordtr 4205 . . . . 5  |-  ( Ord 
A  ->  Tr  A
)
2927, 28syl 14 . . . 4  |-  ( A  e.  On  ->  Tr  A )
30 unisucg 4241 . . . 4  |-  ( A  e.  On  ->  ( Tr  A  <->  U. suc  A  =  A ) )
3129, 30mpbid 145 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
3231adantr 270 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
3326, 32eqtrd 2120 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   U.cuni 3653   Tr wtr 3936   Ord word 4189   Oncon0 4190   suc csuc 4192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198
This theorem is referenced by:  nnpredcl  11845  nnsucpred  11846
  Copyright terms: Public domain W3C validator