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Theorem infxpen 7637
Description: Every infinite ordinal is equinumerous to its cross product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation  R is a well-ordering of  ( On  X.  On ) with the additional property that  R-initial segments of  ( x  X.  x ) (where  x is a limit ordinal) are of cardinality at most  x. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
infxpen  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )

Proof of Theorem infxpen
StepHypRef Expression
1 eqid 2284 . 2  |-  { <. 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 ) ) ) ) }
2 eleq1 2344 . . . . 5  |-  ( s  =  z  ->  (
s  e.  ( On 
X.  On )  <->  z  e.  ( On  X.  On ) ) )
3 eleq1 2344 . . . . 5  |-  ( t  =  w  ->  (
t  e.  ( On 
X.  On )  <->  w  e.  ( On  X.  On ) ) )
42, 3bi2anan9 848 . . . 4  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  <->  ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) ) ) )
5 fveq2 5485 . . . . . . . 8  |-  ( s  =  z  ->  ( 1st `  s )  =  ( 1st `  z
) )
6 fveq2 5485 . . . . . . . 8  |-  ( s  =  z  ->  ( 2nd `  s )  =  ( 2nd `  z
) )
75, 6uneq12d 3331 . . . . . . 7  |-  ( s  =  z  ->  (
( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
87adantr 453 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
9 fveq2 5485 . . . . . . . 8  |-  ( t  =  w  ->  ( 1st `  t )  =  ( 1st `  w
) )
10 fveq2 5485 . . . . . . . 8  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
119, 10uneq12d 3331 . . . . . . 7  |-  ( t  =  w  ->  (
( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
1211adantl 454 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
138, 12eleq12d 2352 . . . . 5  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t )  u.  ( 2nd `  t ) )  <-> 
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
147, 11eqeqan12d 2299 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  <-> 
( ( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
15 breq12 4029 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( s { <. 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 ) ) ) ) } t  <->  z { <. 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 ) ) ) ) } w ) )
1614, 15anbi12d 694 . . . . 5  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( ( ( 1st `  s )  u.  ( 2nd `  s
) )  =  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  /\  s { <. 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
) ) ) ) } t )  <->  ( (
( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  /\  z { <. 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
) ) ) ) } w ) ) )
1713, 16orbi12d 693 . . . 4  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( ( ( 1st `  s )  u.  ( 2nd `  s
) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) )  <->  ( (
( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z { <. 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 ) ) ) ) } w ) ) ) )
184, 17anbi12d 694 . . 3  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  /\  (
( ( 1st `  s
)  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) ) )  <-> 
( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z { <. 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 ) ) ) ) } w ) ) ) ) )
1918cbvopabv 4089 . 2  |-  { <. s ,  t >.  |  ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  s
)  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) ) ) }  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z { <. 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 ) ) ) ) } w ) ) ) }
20 eqid 2284 . 2  |-  ( {
<. s ,  t >.  |  ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  /\  (
( ( 1st `  s
)  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) ) ) }  i^i  ( ( a  X.  a )  X.  ( a  X.  a ) ) )  =  ( { <. s ,  t >.  |  ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  s
)  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) ) ) }  i^i  ( ( a  X.  a )  X.  ( a  X.  a ) ) )
21 biid 229 . 2  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  <->  ( (
a  e.  On  /\  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m ) 
~~  m ) )  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a )
) )
22 eqid 2284 . 2  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
23 eqid 2284 . 2  |- OrdIso ( ( { <. s ,  t
>.  |  ( (
s  e.  ( On 
X.  On )  /\  t  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t )  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s
) )  =  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  /\  s { <. 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
) ) ) ) } t ) ) ) }  i^i  (
( a  X.  a
)  X.  ( a  X.  a ) ) ) ,  ( a  X.  a ) )  = OrdIso ( ( {
<. s ,  t >.  |  ( ( s  e.  ( On  X.  On )  /\  t  e.  ( On  X.  On ) )  /\  (
( ( 1st `  s
)  u.  ( 2nd `  s ) )  e.  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  \/  ( ( ( 1st `  s )  u.  ( 2nd `  s ) )  =  ( ( 1st `  t )  u.  ( 2nd `  t ) )  /\  s { <. 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 ) ) ) ) } t ) ) ) }  i^i  ( ( a  X.  a )  X.  ( a  X.  a ) ) ) ,  ( a  X.  a ) )
241, 19, 20, 21, 22, 23infxpenlem 7636 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1628    e. wcel 1688   A.wral 2544    u. cun 3151    i^i cin 3152    C_ wss 3153   class class class wbr 4024   {copab 4077   Oncon0 4391   omcom 4655    X. cxp 4686   ` cfv 5221   1stc1st 6081   2ndc2nd 6082    ~~ cen 6855    ~< csdm 6857  OrdIsocoi 7219
This theorem is referenced by:  xpomen  7638  infxpidm2  7639  alephreg  8199  cfpwsdom  8201  inar1  8392
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-card 7567
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