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Theorem infxpen 7610
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 2258 . 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 2318 . . . . 5  |-  ( s  =  z  ->  (
s  e.  ( On 
X.  On )  <->  z  e.  ( On  X.  On ) ) )
3 eleq1 2318 . . . . 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 5458 . . . . . . . 8  |-  ( s  =  z  ->  ( 1st `  s )  =  ( 1st `  z
) )
6 fveq2 5458 . . . . . . . 8  |-  ( s  =  z  ->  ( 2nd `  s )  =  ( 2nd `  z
) )
75, 6uneq12d 3305 . . . . . . 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 5458 . . . . . . . 8  |-  ( t  =  w  ->  ( 1st `  t )  =  ( 1st `  w
) )
10 fveq2 5458 . . . . . . . 8  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
119, 10uneq12d 3305 . . . . . . 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 2326 . . . . 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 2273 . . . . . 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 4002 . . . . . 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 4062 . 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 2258 . 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 2258 . 2  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
23 eqid 2258 . 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 7609 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 1619    e. wcel 1621   A.wral 2518    u. cun 3125    i^i cin 3126    C_ wss 3127   class class class wbr 3997   {copab 4050   Oncon0 4364   omcom 4628    X. cxp 4659   ` cfv 4673   1stc1st 6054   2ndc2nd 6055    ~~ cen 6828    ~< csdm 6830  OrdIsocoi 7192
This theorem is referenced by:  xpomen  7611  infxpidm2  7612  alephreg  8172  cfpwsdom  8174  inar1  8365
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-oi 7193  df-card 7540
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