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Theorem wexp 6887
Description: A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
Hypothesis
Ref Expression
wexp.1  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
Assertion
Ref Expression
wexp  |-  ( ( R  We  A  /\  S  We  B )  ->  T  We  ( A  X.  B ) )
Distinct variable groups:    x, A, y    x, B, y    x, R, y    x, S, y
Allowed substitution hints:    T( x, y)

Proof of Theorem wexp
StepHypRef Expression
1 wefr 4858 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 wefr 4858 . . 3  |-  ( S  We  B  ->  S  Fr  B )
3 wexp.1 . . . 4  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
43frxp 6883 . . 3  |-  ( ( R  Fr  A  /\  S  Fr  B )  ->  T  Fr  ( A  X.  B ) )
51, 2, 4syl2an 475 . 2  |-  ( ( R  We  A  /\  S  We  B )  ->  T  Fr  ( A  X.  B ) )
6 weso 4859 . . 3  |-  ( R  We  A  ->  R  Or  A )
7 weso 4859 . . 3  |-  ( S  We  B  ->  S  Or  B )
83soxp 6886 . . 3  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Or  ( A  X.  B ) )
96, 7, 8syl2an 475 . 2  |-  ( ( R  We  A  /\  S  We  B )  ->  T  Or  ( A  X.  B ) )
10 df-we 4829 . 2  |-  ( T  We  ( A  X.  B )  <->  ( T  Fr  ( A  X.  B
)  /\  T  Or  ( A  X.  B
) ) )
115, 9, 10sylanbrc 662 1  |-  ( ( R  We  A  /\  S  We  B )  ->  T  We  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   {copab 4496    Or wor 4788    Fr wfr 4824    We wwe 4826    X. cxp 4986   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  fnwelem  6888  leweon  8380
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