Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsuc2 Structured version   Unicode version

Theorem limsuc2 35328
Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Assertion
Ref Expression
limsuc2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )

Proof of Theorem limsuc2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordunisuc2 6661 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
21biimpa 482 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 5474 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2471 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspccva 3158 . . . 4  |-  ( ( A. x  e.  A  suc  x  e.  A  /\  B  e.  A )  ->  suc  B  e.  A
)
62, 5sylan 469 . . 3  |-  ( ( ( Ord  A  /\  A  =  U. A )  /\  B  e.  A
)  ->  suc  B  e.  A )
76ex 432 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  ->  suc  B  e.  A
) )
8 ordtr 5423 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 5493 . . . . 5  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 432 . . . 4  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
118, 10syl 17 . . 3  |-  ( Ord 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
1211adantr 463 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( suc  B  e.  A  ->  B  e.  A
) )
137, 12impbid 191 1  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   U.cuni 4190   Tr wtr 4488   Ord word 5408   suc csuc 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412  df-on 5413  df-suc 5415
This theorem is referenced by:  aomclem4  35345
  Copyright terms: Public domain W3C validator