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Theorem infxpidm 8066
Description: The cross 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 7526. (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 6755 . . . 4  |-  Rel  ~<_
21brrelex2i 4637 . . 3  |-  ( om  ~<_  A  ->  A  e.  _V )
3 numth3 7981 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
42, 3syl 17 . 2  |-  ( om  ~<_  A  ->  A  e.  dom  card )
5 infxpidm2 7528 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
64, 5mpancom 653 1  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2727   class class class wbr 3920   omcom 4547    X. cxp 4578   dom cdm 4580    ~~ cen 6746    ~<_ cdom 6747   cardccrd 7452
This theorem is referenced by:  inar1  8277
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-ac2 7973
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-ac 7627
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