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Theorem wexp 6791
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 4813 . . 3  |-  ( R  We  A  ->  R  Fr  A )
2 wefr 4813 . . 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 6787 . . 3  |-  ( ( R  Fr  A  /\  S  Fr  B )  ->  T  Fr  ( A  X.  B ) )
51, 2, 4syl2an 477 . 2  |-  ( ( R  We  A  /\  S  We  B )  ->  T  Fr  ( A  X.  B ) )
6 weso 4814 . . 3  |-  ( R  We  A  ->  R  Or  A )
7 weso 4814 . . 3  |-  ( S  We  B  ->  S  Or  B )
83soxp 6790 . . 3  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Or  ( A  X.  B ) )
96, 7, 8syl2an 477 . 2  |-  ( ( R  We  A  /\  S  We  B )  ->  T  Or  ( A  X.  B ) )
10 df-we 4784 . 2  |-  ( T  We  ( A  X.  B )  <->  ( T  Fr  ( A  X.  B
)  /\  T  Or  ( A  X.  B
) ) )
115, 9, 10sylanbrc 664 1  |-  ( ( R  We  A  /\  S  We  B )  ->  T  We  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4395   {copab 4452    Or wor 4743    Fr wfr 4779    We wwe 4781    X. cxp 4941   ` cfv 5521   1stc1st 6680   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-1st 6682  df-2nd 6683
This theorem is referenced by:  fnwelem  6792  leweon  8284
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