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Theorem infxpen 7880
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 2430 . 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 2490 . . . . 5  |-  ( s  =  z  ->  (
s  e.  ( On 
X.  On )  <->  z  e.  ( On  X.  On ) ) )
3 eleq1 2490 . . . . 5  |-  ( t  =  w  ->  (
t  e.  ( On 
X.  On )  <->  w  e.  ( On  X.  On ) ) )
42, 3bi2anan9 844 . . . 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 5714 . . . . . . . 8  |-  ( s  =  z  ->  ( 1st `  s )  =  ( 1st `  z
) )
6 fveq2 5714 . . . . . . . 8  |-  ( s  =  z  ->  ( 2nd `  s )  =  ( 2nd `  z
) )
75, 6uneq12d 3489 . . . . . . 7  |-  ( s  =  z  ->  (
( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
87adantr 452 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  s
)  u.  ( 2nd `  s ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
9 fveq2 5714 . . . . . . . 8  |-  ( t  =  w  ->  ( 1st `  t )  =  ( 1st `  w
) )
10 fveq2 5714 . . . . . . . 8  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
119, 10uneq12d 3489 . . . . . . 7  |-  ( t  =  w  ->  (
( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
1211adantl 453 . . . . . 6  |-  ( ( s  =  z  /\  t  =  w )  ->  ( ( 1st `  t
)  u.  ( 2nd `  t ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
138, 12eleq12d 2498 . . . . 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 2445 . . . . . 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 4204 . . . . . 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 692 . . . . 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 691 . . . 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 692 . . 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 4264 . 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 2430 . 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 228 . 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 2430 . 2  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
23 eqid 2430 . 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 7879 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2692    u. cun 3305    i^i cin 3306    C_ wss 3307   class class class wbr 4199   {copab 4252   Oncon0 4568   omcom 4831    X. cxp 4862   ` cfv 5440   1stc1st 6333   2ndc2nd 6334    ~~ cen 7092    ~< csdm 7094  OrdIsocoi 7462
This theorem is referenced by:  xpomen  7881  infxpidm2  7882  alephreg  8441  cfpwsdom  8443  inar1  8634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-oi 7463  df-card 7810
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