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Theorem xp2dju 9605
 Description: Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xp2dju (2o × 𝐴) = (𝐴𝐴)

Proof of Theorem xp2dju
StepHypRef Expression
1 xpundir 5589 . 2 (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴))
2 df2o3 8118 . . . 4 2o = {∅, 1o}
3 df-pr 4531 . . . 4 {∅, 1o} = ({∅} ∪ {1o})
42, 3eqtri 2821 . . 3 2o = ({∅} ∪ {1o})
54xpeq1i 5549 . 2 (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴)
6 df-dju 9332 . 2 (𝐴𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴))
71, 5, 63eqtr4i 2831 1 (2o × 𝐴) = (𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3881  ∅c0 4246  {csn 4528  {cpr 4530   × cxp 5521  1oc1o 8096  2oc2o 8097   ⊔ cdju 9329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-pr 4531  df-opab 5097  df-xp 5529  df-suc 6172  df-1o 8103  df-2o 8104  df-dju 9332 This theorem is referenced by:  pwdju1  9619  unctb  9634  infdjuabs  9635  ackbij1lem5  9653  fin56  9822
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