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Theorem ordelordALT 28301
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4414 using the Axiom of Regularity indirectly through dford2 7321. 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 28301 is ordelordALTVD 28643 without virtual deductions and was automatically derived from ordelordALTVD 28643 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 4406 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
21adantr 451 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  Tr  A )
3 dford2 7321 . . . . . 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 2569 . . . 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 28299 . . 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 4122 . . . 4  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
132, 9, 12sylc 56 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
14 ssralv2 28294 . . . 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 7321 . 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 1684   A.wral 2543    C_ wss 3152   Tr wtr 4113   Ord word 4391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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