MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  leweon Unicode version

Theorem leweon 7523
Description: Lexicographical order is a well-ordering of  On  X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7524, 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 4466 . 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 5391 . . . . . . . 8  |-  ( 1st `  y )  e.  _V
43epelc 4200 . . . . . . 7  |-  ( ( 1st `  x )  _E  ( 1st `  y
)  <->  ( 1st `  x
)  e.  ( 1st `  y ) )
5 fvex 5391 . . . . . . . . 9  |-  ( 2nd `  y )  e.  _V
65epelc 4200 . . . . . . . 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 3980 . . . 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 2276 . . 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 6081 . 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 3920   {copab 3973    _E cep 4196    We wwe 4244   Oncon0 4285    X. cxp 4578   ` cfv 4592   1stc1st 5972   2ndc2nd 5973
This theorem is referenced by:  r0weon  7524
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-1st 5974  df-2nd 5975
  Copyright terms: Public domain W3C validator