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Theorem leweon 7931
Description: Lexicographical order is a well-ordering of  On  X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7932, 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 4799 . 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 5773 . . . . . . . 8  |-  ( 1st `  y )  e.  _V
43epelc 4531 . . . . . . 7  |-  ( ( 1st `  x )  _E  ( 1st `  y
)  <->  ( 1st `  x
)  e.  ( 1st `  y ) )
5 fvex 5773 . . . . . . . . 9  |-  ( 2nd `  y )  e.  _V
65epelc 4531 . . . . . . . 8  |-  ( ( 2nd `  x )  _E  ( 2nd `  y
)  <->  ( 2nd `  x
)  e.  ( 2nd `  y ) )
76anbi2i 677 . . . . . . 7  |-  ( ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  _E  ( 2nd `  y
) )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) )
84, 7orbi12i 509 . . . . . 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 677 . . . . 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 4303 . . . 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 2466 . . 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 6496 . 2  |-  ( (  _E  We  On  /\  _E  We  On )  ->  L  We  ( On  X.  On ) )
131, 1, 12mp2an 655 1  |-  L  We  ( On  X.  On )
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360    = wceq 1654    e. wcel 1728   class class class wbr 4243   {copab 4296    _E cep 4527    We wwe 4575   Oncon0 4616    X. cxp 4911   ` cfv 5489   1stc1st 6383   2ndc2nd 6384
This theorem is referenced by:  r0weon  7932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fv 5497  df-1st 6385  df-2nd 6386
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