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Theorem infxpidm 9962
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 9418. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
Assertion
Ref Expression
infxpidm (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)

Proof of Theorem infxpidm
StepHypRef Expression
1 reldom 8493 . . . 4 Rel ≼
21brrelex2i 5585 . . 3 (ω ≼ 𝐴𝐴 ∈ V)
3 numth3 9870 . . 3 (𝐴 ∈ V → 𝐴 ∈ dom card)
42, 3syl 17 . 2 (ω ≼ 𝐴𝐴 ∈ dom card)
5 infxpidm2 9421 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
64, 5mpancom 686 1 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Vcvv 3473   class class class wbr 5042   × cxp 5529  dom cdm 5531  ωcom 7558  cen 8484  cdom 8485  cardccrd 9342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-ac2 9863
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-oi 8952  df-card 9346  df-ac 9520
This theorem is referenced by:  unirnfdomd  9967  inar1  10175
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