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Theorem ordelordALT 28696
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4606 using the Axiom of Regularity indirectly through dford2 7578. 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 28696 is ordelordALTVD 29053 without virtual deductions and was automatically derived from ordelordALTVD 29053 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 4598 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
21adantr 453 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  A )
3 dford2 7578 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
43simprbi 452 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
54adantr 453 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 3orcomb 947 . . . . 5  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  y  \/  y  e.  x  \/  x  =  y
) )
762ralbii 2733 . . . 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 190 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9 simpr 449 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  A )
10 tratrb 28694 . . 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 1185 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  B )
12 trss 4314 . . . 4  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
132, 9, 12sylc 59 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
14 ssralv2 28689 . . . 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 425 . . 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 60 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
17 dford2 7578 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
1811, 16, 17sylanbrc 647 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    \/ w3o 936    e. wcel 1726   A.wral 2707    C_ wss 3322   Tr wtr 4305   Ord word 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704  ax-reg 7563
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587
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