MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordsucuniel Unicode version

Theorem ordsucuniel 4745
Description: Given an element  A of the union of an ordinal  B,  suc  A is an element of  B itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
ordsucuniel  |-  ( Ord 
B  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )

Proof of Theorem ordsucuniel
StepHypRef Expression
1 orduni 4715 . . 3  |-  ( Ord 
B  ->  Ord  U. B
)
2 ordelord 4545 . . . 4  |-  ( ( Ord  U. B  /\  A  e.  U. B )  ->  Ord  A )
32ex 424 . . 3  |-  ( Ord  U. B  ->  ( A  e.  U. B  ->  Ord  A ) )
41, 3syl 16 . 2  |-  ( Ord 
B  ->  ( A  e.  U. B  ->  Ord  A ) )
5 ordelord 4545 . . . 4  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  suc  A )
6 ordsuc 4735 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
75, 6sylibr 204 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  A )
87ex 424 . 2  |-  ( Ord 
B  ->  ( suc  A  e.  B  ->  Ord  A ) )
9 ordsson 4711 . . . . . 6  |-  ( Ord 
B  ->  B  C_  On )
10 ordunisssuc 4625 . . . . . 6  |-  ( ( B  C_  On  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
119, 10sylan 458 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
12 ordtri1 4556 . . . . . 6  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
131, 12sylan 458 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
14 ordtri1 4556 . . . . . 6  |-  ( ( Ord  B  /\  Ord  suc 
A )  ->  ( B  C_  suc  A  <->  -.  suc  A  e.  B ) )
156, 14sylan2b 462 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_ 
suc  A  <->  -.  suc  A  e.  B ) )
1611, 13, 153bitr3d 275 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( -.  A  e.  U. B  <->  -.  suc  A  e.  B ) )
1716con4bid 285 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
1817ex 424 . 2  |-  ( Ord 
B  ->  ( Ord  A  ->  ( A  e. 
U. B  <->  suc  A  e.  B ) ) )
194, 8, 18pm5.21ndd 344 1  |-  ( Ord 
B  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    C_ wss 3264   U.cuni 3958   Ord word 4522   Oncon0 4523   suc csuc 4525
This theorem is referenced by:  dfac12lem1  7957  dfac12lem2  7958  nofulllem5  25385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529
  Copyright terms: Public domain W3C validator