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Theorem leweon 7572
Description: Lexicographical order is a well-ordering of  On  X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7573, 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 4512 . 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 5437 . . . . . . . 8  |-  ( 1st `  y )  e.  _V
43epelc 4244 . . . . . . 7  |-  ( ( 1st `  x )  _E  ( 1st `  y
)  <->  ( 1st `  x
)  e.  ( 1st `  y ) )
5 fvex 5437 . . . . . . . . 9  |-  ( 2nd `  y )  e.  _V
65epelc 4244 . . . . . . . 8  |-  ( ( 2nd `  x )  _E  ( 2nd `  y
)  <->  ( 2nd `  x
)  e.  ( 2nd `  y ) )
76anbi2i 678 . . . . . . 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 678 . . . . 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 4023 . . . 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 2279 . . 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 6128 . 2  |-  ( (  _E  We  On  /\  _E  We  On )  ->  L  We  ( On  X.  On ) )
131, 1, 12mp2an 656 1  |-  L  We  ( On  X.  On )
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3963   {copab 4016    _E cep 4240    We wwe 4288   Oncon0 4329    X. cxp 4624   ` cfv 4638   1stc1st 6019   2ndc2nd 6020
This theorem is referenced by:  r0weon  7573
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-1st 6021  df-2nd 6022
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