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Theorem ordsucuniel 4796
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 4766 . . 3  |-  ( Ord 
B  ->  Ord  U. B
)
2 ordelord 4595 . . . 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 4595 . . . 4  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  suc  A )
6 ordsuc 4786 . . . 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 4762 . . . . . 6  |-  ( Ord 
B  ->  B  C_  On )
10 ordunisssuc 4676 . . . . . 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 4606 . . . . . 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 4606 . . . . . 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 1725    C_ wss 3312   U.cuni 4007   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  dfac12lem1  8015  dfac12lem2  8016  nofulllem5  25653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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