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Theorem infxpen 7658
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
Dummy variables  m  a  s  t  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . 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 2356 . . . . 5  |-  ( s  =  z  ->  (
s  e.  ( On 
X.  On )  <->  z  e.  ( On  X.  On ) ) )
3 eleq1 2356 . . . . 5  |-  ( t  =  w  ->  (
t  e.  ( On 
X.  On )  <->  w  e.  ( On  X.  On ) ) )
42, 3bi2anan9 843 . . . 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 5541 . . . . . . . 8  |-  ( s  =  z  ->  ( 1st `  s )  =  ( 1st `  z
) )
6 fveq2 5541 . . . . . . . 8  |-  ( s  =  z  ->  ( 2nd `  s )  =  ( 2nd `  z
) )
75, 6uneq12d 3343 . . . . . . 7  |-  ( s  =  z  ->  (
( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
87adantr 451 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
9 fveq2 5541 . . . . . . . 8  |-  ( t  =  w  ->  ( 1st `  t )  =  ( 1st `  w
) )
10 fveq2 5541 . . . . . . . 8  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
119, 10uneq12d 3343 . . . . . . 7  |-  ( t  =  w  ->  (
( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
1211adantl 452 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
138, 12eleq12d 2364 . . . . 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 2311 . . . . . 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 4044 . . . . . 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 691 . . . . 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 690 . . . 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 691 . . 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 4104 . 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 2296 . 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 227 . 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 2296 . 2  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
23 eqid 2296 . 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 7657 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163    i^i cin 3164    C_ wss 3165   class class class wbr 4039   {copab 4092   Oncon0 4408   omcom 4672    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137    ~~ cen 6876    ~< csdm 6878  OrdIsocoi 7240
This theorem is referenced by:  xpomen  7659  infxpidm2  7660  alephreg  8220  cfpwsdom  8222  inar1  8413
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  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-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588
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