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Theorem ordelordALT 27795
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4451 using the Axiom of Regularity indirectly through dford2 7366. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that  _E  Fr  A because this is inferred by the Axiom of Regularity. ordelordALT 27795 is ordelordALTVD 28154 without virtual deductions and was automatically derived from ordelordALTVD 28154 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ordelordALT  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )

Proof of Theorem ordelordALT
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtr 4443 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
21adantr 451 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  A )
3 dford2 7366 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
43simprbi 450 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
54adantr 451 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 3orcomb 944 . . . . 5  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  y  \/  y  e.  x  \/  x  =  y
) )
762ralbii 2603 . . . 4  |-  ( A. x  e.  A  A. y  e.  A  (
x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y
) )
85, 7sylib 188 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9 simpr 447 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  A )
10 tratrb 27793 . . 3  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A  (
x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A
)  ->  Tr  B
)
112, 8, 9, 10syl3anc 1182 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  B )
12 trss 4159 . . . 4  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
132, 9, 12sylc 56 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
14 ssralv2 27788 . . . 4  |-  ( ( B  C_  A  /\  B  C_  A )  -> 
( A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) )
1514ex 423 . . 3  |-  ( B 
C_  A  ->  ( B  C_  A  ->  ( A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
)  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) ) )
1613, 13, 5, 15syl3c 57 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
17 dford2 7366 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
1811, 16, 17sylanbrc 645 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    e. wcel 1701   A.wral 2577    C_ wss 3186   Tr wtr 4150   Ord word 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549  ax-reg 7351
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432
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