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Theorem infxpidm2 8426
Description: The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 8969. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxpidm2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )

Proof of Theorem infxpidm2
StepHypRef Expression
1 cardid2 8366 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7604 . . . . 5  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 xpen 7718 . . . . 5  |-  ( ( A  ~~  ( card `  A )  /\  A  ~~  ( card `  A
) )  ->  ( A  X.  A )  ~~  ( ( card `  A
)  X.  ( card `  A ) ) )
42, 2, 3syl2anc 659 . . . 4  |-  ( A  e.  dom  card  ->  ( A  X.  A ) 
~~  ( ( card `  A )  X.  ( card `  A ) ) )
54adantr 463 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) ) )
6 cardon 8357 . . . 4  |-  ( card `  A )  e.  On
7 cardom 8399 . . . . 5  |-  ( card `  om )  =  om
8 omelon 8096 . . . . . . . 8  |-  om  e.  On
9 onenon 8362 . . . . . . . 8  |-  ( om  e.  On  ->  om  e.  dom  card )
108, 9ax-mp 5 . . . . . . 7  |-  om  e.  dom  card
11 carddom2 8390 . . . . . . 7  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
1210, 11mpan 668 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
1312biimpar 483 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
147, 13syl5eqssr 3487 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
15 infxpen 8424 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  om  C_  ( card `  A
) )  ->  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )
166, 14, 15sylancr 661 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  A
)  X.  ( card `  A ) )  ~~  ( card `  A )
)
17 entr 7605 . . 3  |-  ( ( ( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) )  /\  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )  ->  ( A  X.  A )  ~~  ( card `  A ) )
185, 16, 17syl2anc 659 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( card `  A ) )
191adantr 463 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
20 entr 7605 . 2  |-  ( ( ( A  X.  A
)  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  ( A  X.  A )  ~~  A )
2118, 19, 20syl2anc 659 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1842    C_ wss 3414   class class class wbr 4395    X. cxp 4821   dom cdm 4823   Oncon0 5410   ` cfv 5569   omcom 6683    ~~ cen 7551    ~<_ cdom 7552   cardccrd 8348
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-card 8352
This theorem is referenced by:  infpwfien  8475  mappwen  8525  infcdaabs  8618  infxpdom  8623  fin67  8807  infxpidm  8969  ttac  35340
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