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Theorem ordelordALT 36968
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5452 using the Axiom of Regularity indirectly through dford2 8143. 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 36968 is ordelordALTVD 37327 without virtual deductions and was automatically derived from ordelordALTVD 37327 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 5444 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
21adantr 472 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  A )
3 dford2 8143 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
43simprbi 471 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
54adantr 472 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 3orcomb 1017 . . . . 5  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <-> 
( x  e.  y  \/  y  e.  x  \/  x  =  y
) )
762ralbii 2824 . . . 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 201 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9 simpr 468 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  A )
10 tratrb 36967 . . 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 1292 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  B )
12 trss 4499 . . . 4  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
132, 9, 12sylc 61 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
14 ssralv2 36958 . . . 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 441 . . 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 62 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
17 dford2 8143 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  A. y  e.  B  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
1811, 16, 17sylanbrc 677 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    \/ w3o 1006    e. wcel 1904   A.wral 2756    C_ wss 3390   Tr wtr 4490   Ord word 5429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602  ax-reg 8125
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433
This theorem is referenced by: (None)
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