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Theorem infxpen 8383
Description: Every infinite ordinal is equinumerous to its Cartesian 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
Dummy variables  m  a  s  t  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . 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 2526 . . . . 5  |-  ( s  =  z  ->  (
s  e.  ( On 
X.  On )  <->  z  e.  ( On  X.  On ) ) )
3 eleq1 2526 . . . . 5  |-  ( t  =  w  ->  (
t  e.  ( On 
X.  On )  <->  w  e.  ( On  X.  On ) ) )
42, 3bi2anan9 871 . . . 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 5848 . . . . . . . 8  |-  ( s  =  z  ->  ( 1st `  s )  =  ( 1st `  z
) )
6 fveq2 5848 . . . . . . . 8  |-  ( s  =  z  ->  ( 2nd `  s )  =  ( 2nd `  z
) )
75, 6uneq12d 3645 . . . . . . 7  |-  ( s  =  z  ->  (
( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
87adantr 463 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
9 fveq2 5848 . . . . . . . 8  |-  ( t  =  w  ->  ( 1st `  t )  =  ( 1st `  w
) )
10 fveq2 5848 . . . . . . . 8  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
119, 10uneq12d 3645 . . . . . . 7  |-  ( t  =  w  ->  (
( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
1211adantl 464 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
138, 12eleq12d 2536 . . . . 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 2477 . . . . . 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 4444 . . . . . 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 708 . . . . 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 707 . . . 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 708 . . 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 4508 . 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 2454 . 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 236 . 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 2454 . 2  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
23 eqid 2454 . 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 8382 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    u. cun 3459    i^i cin 3460    C_ wss 3461   class class class wbr 4439   {copab 4496   Oncon0 4867    X. cxp 4986   ` cfv 5570   omcom 6673   1stc1st 6771   2ndc2nd 6772    ~~ cen 7506    ~< csdm 7508  OrdIsocoi 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311
This theorem is referenced by:  xpomen  8384  infxpidm2  8385  alephreg  8948  cfpwsdom  8950  inar1  9142
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