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Theorem ordsucuniel 4631
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 4601 . . 3  |-  ( Ord 
B  ->  Ord  U. B
)
2 ordelord 4430 . . . 4  |-  ( ( Ord  U. B  /\  A  e.  U. B )  ->  Ord  A )
32ex 423 . . 3  |-  ( Ord  U. B  ->  ( A  e.  U. B  ->  Ord  A ) )
41, 3syl 15 . 2  |-  ( Ord 
B  ->  ( A  e.  U. B  ->  Ord  A ) )
5 ordelord 4430 . . . 4  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  suc  A )
6 ordsuc 4621 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
75, 6sylibr 203 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  B )  ->  Ord  A )
87ex 423 . 2  |-  ( Ord 
B  ->  ( suc  A  e.  B  ->  Ord  A ) )
9 ordsson 4597 . . . . . 6  |-  ( Ord 
B  ->  B  C_  On )
10 ordunisssuc 4511 . . . . . 6  |-  ( ( B  C_  On  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
119, 10sylan 457 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  B  C_  suc  A ) )
12 ordtri1 4441 . . . . . 6  |-  ( ( Ord  U. B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
131, 12sylan 457 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( U. B  C_  A  <->  -.  A  e.  U. B ) )
14 ordtri1 4441 . . . . . 6  |-  ( ( Ord  B  /\  Ord  suc 
A )  ->  ( B  C_  suc  A  <->  -.  suc  A  e.  B ) )
156, 14sylan2b 461 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_ 
suc  A  <->  -.  suc  A  e.  B ) )
1611, 13, 153bitr3d 274 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( -.  A  e.  U. B  <->  -.  suc  A  e.  B ) )
1716con4bid 284 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
1817ex 423 . 2  |-  ( Ord 
B  ->  ( Ord  A  ->  ( A  e. 
U. B  <->  suc  A  e.  B ) ) )
194, 8, 18pm5.21ndd 343 1  |-  ( Ord 
B  ->  ( A  e.  U. B  <->  suc  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    C_ wss 3165   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  dfac12lem1  7785  dfac12lem2  7786  nofulllem5  24431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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