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Theorem leweon 7639
Description: Lexicographical order is a well-ordering of  On  X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7640, this order is not set-like, as the preimage of  <. 1o ,  (/) >. is the proper class  ( { (/) }  X.  On ). (Contributed by Mario Carneiro, 9-Mar-2013.)
Hypothesis
Ref Expression
leweon.1  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
Assertion
Ref Expression
leweon  |-  L  We  ( On  X.  On )
Distinct variable group:    x, y
Allowed substitution hints:    L( x, y)

Proof of Theorem leweon
StepHypRef Expression
1 epweon 4575 . 2  |-  _E  We  On
2 leweon.1 . . . 4  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
3 fvex 5539 . . . . . . . 8  |-  ( 1st `  y )  e.  _V
43epelc 4307 . . . . . . 7  |-  ( ( 1st `  x )  _E  ( 1st `  y
)  <->  ( 1st `  x
)  e.  ( 1st `  y ) )
5 fvex 5539 . . . . . . . . 9  |-  ( 2nd `  y )  e.  _V
65epelc 4307 . . . . . . . 8  |-  ( ( 2nd `  x )  _E  ( 2nd `  y
)  <->  ( 2nd `  x
)  e.  ( 2nd `  y ) )
76anbi2i 675 . . . . . . 7  |-  ( ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  _E  ( 2nd `  y
) )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) )
84, 7orbi12i 507 . . . . . 6  |-  ( ( ( 1st `  x
)  _E  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  _E  ( 2nd `  y
) ) )  <->  ( ( 1st `  x )  e.  ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) )
98anbi2i 675 . . . . 5  |-  ( ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x )  _E  ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  _E  ( 2nd `  y ) ) ) )  <->  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) )
109opabbii 4083 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x )  _E  ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  _E  ( 2nd `  y ) ) ) ) }  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
112, 10eqtr4i 2306 . . 3  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  _E  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  _E  ( 2nd `  y
) ) ) ) }
1211wexp 6229 . 2  |-  ( (  _E  We  On  /\  _E  We  On )  ->  L  We  ( On  X.  On ) )
131, 1, 12mp2an 653 1  |-  L  We  ( On  X.  On )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   {copab 4076    _E cep 4303    We wwe 4351   Oncon0 4392    X. cxp 4687   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  r0weon  7640
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-pow 4188  ax-pr 4214  ax-un 4512
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-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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