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Theorem ordelordALT 28477
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4595 using the Axiom of Regularity indirectly through dford2 7564. 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 28477 is ordelordALTVD 28833 without virtual deductions and was automatically derived from ordelordALTVD 28833 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 4587 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
21adantr 452 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  A )
3 dford2 7564 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
43simprbi 451 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
54adantr 452 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 3orcomb 946 . . . . 5  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  y  \/  y  e.  x  \/  x  =  y
) )
762ralbii 2723 . . . 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 189 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9 simpr 448 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  A )
10 tratrb 28475 . . 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 1184 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  B )
12 trss 4303 . . . 4  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
132, 9, 12sylc 58 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
14 ssralv2 28470 . . . 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 424 . . 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 59 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
17 dford2 7564 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
1811, 16, 17sylanbrc 646 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    e. wcel 1725   A.wral 2697    C_ wss 3312   Tr wtr 4294   Ord word 4572
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 4692  ax-reg 7549
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-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
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