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Theorem ordsucuniel 6632
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 6602 . . 3  |-  ( Ord 
B  ->  Ord  U. B
)
2 ordelord 4895 . . . 4  |-  ( ( Ord  U. B  /\  A  e.  U. B )  ->  Ord  A )
32ex 434 . . 3  |-  ( Ord  U. B  ->  ( A  e.  U. B  ->  Ord  A ) )
41, 3syl 16 . 2  |-  ( Ord 
B  ->  ( A  e.  U. B  ->  Ord  A ) )
5 ordelord 4895 . . . 4  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  suc  A )
6 ordsuc 6622 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
75, 6sylibr 212 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  A )
87ex 434 . 2  |-  ( Ord 
B  ->  ( suc  A  e.  B  ->  Ord  A ) )
9 ordsson 6598 . . . . . 6  |-  ( Ord 
B  ->  B  C_  On )
10 ordunisssuc 4975 . . . . . 6  |-  ( ( B  C_  On  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
119, 10sylan 471 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
12 ordtri1 4906 . . . . . 6  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
131, 12sylan 471 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
14 ordtri1 4906 . . . . . 6  |-  ( ( Ord  B  /\  Ord  suc 
A )  ->  ( B  C_  suc  A  <->  -.  suc  A  e.  B ) )
156, 14sylan2b 475 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_ 
suc  A  <->  -.  suc  A  e.  B ) )
1611, 13, 153bitr3d 283 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( -.  A  e.  U. B  <->  -.  suc  A  e.  B ) )
1716con4bid 293 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
1817ex 434 . 2  |-  ( Ord 
B  ->  ( Ord  A  ->  ( A  e. 
U. B  <->  suc  A  e.  B ) ) )
194, 8, 18pm5.21ndd 354 1  |-  ( Ord 
B  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762    C_ wss 3471   U.cuni 4240   Ord word 4872   Oncon0 4873   suc csuc 4875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-tr 4536  df-eprel 4786  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-suc 4879
This theorem is referenced by:  dfac12lem1  8514  dfac12lem2  8515  nofulllem5  29031
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