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Theorem leweon 7655
Description: Lexicographical order is a well-ordering of  On  X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7656, 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 4591 . 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 5555 . . . . . . . 8  |-  ( 1st `  y )  e.  _V
43epelc 4323 . . . . . . 7  |-  ( ( 1st `  x )  _E  ( 1st `  y
)  <->  ( 1st `  x
)  e.  ( 1st `  y ) )
5 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  y )  e.  _V
65epelc 4323 . . . . . . . 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 4099 . . . 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 2319 . . 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 6245 . 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 1632    e. wcel 1696   class class class wbr 4039   {copab 4092    _E cep 4319    We wwe 4367   Oncon0 4408    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  r0weon  7656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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