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Theorem infxpidm 8889
Description: The Cartesian product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 8344. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
Assertion
Ref Expression
infxpidm  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~~  A
)

Proof of Theorem infxpidm
StepHypRef Expression
1 reldom 7480 . . . 4  |-  Rel  ~<_
21brrelex2i 4984 . . 3  |-  ( om  ~<_  A  ->  A  e.  _V )
3 numth3 8802 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
42, 3syl 17 . 2  |-  ( om  ~<_  A  ->  A  e.  dom  card )
5 infxpidm2 8346 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
64, 5mpancom 667 1  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~~  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   _Vcvv 3058   class class class wbr 4394    X. cxp 4940   dom cdm 4942   omcom 6638    ~~ cen 7471    ~<_ cdom 7472   cardccrd 8268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-ac2 8795
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-oi 7889  df-card 8272  df-ac 8449
This theorem is referenced by:  unirnfdomd  8894  inar1  9103
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