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Theorem xp2dju 10120
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 5705 . 2 (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴))
2 df2o3 8424 . . . 4 2o = {∅, 1o}
3 df-pr 4593 . . . 4 {∅, 1o} = ({∅} ∪ {1o})
42, 3eqtri 2761 . . 3 2o = ({∅} ∪ {1o})
54xpeq1i 5663 . 2 (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴)
6 df-dju 9845 . 2 (𝐴𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴))
71, 5, 63eqtr4i 2771 1 (2o × 𝐴) = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3912  c0 4286  {csn 4590  {cpr 4592   × cxp 5635  1oc1o 8409  2oc2o 8410  cdju 9842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-dif 3917  df-un 3919  df-nul 4287  df-pr 4593  df-opab 5172  df-xp 5643  df-suc 6327  df-1o 8416  df-2o 8417  df-dju 9845
This theorem is referenced by:  pwdju1  10134  unctb  10149  infdjuabs  10150  ackbij1lem5  10168  fin56  10337
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