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Theorem xp2cda 9040
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
xp2cda (𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Proof of Theorem xp2cda
StepHypRef Expression
1 cdaval 9030 . . 3 ((𝐴𝑉𝐴𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
21anidms 678 . 2 (𝐴𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3 df2o3 7618 . . . . 5 2𝑜 = {∅, 1𝑜}
4 df-pr 4213 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
53, 4eqtri 2673 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
65xpeq2i 5170 . . 3 (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7 xpundi 5205 . . 3 (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
86, 7eqtri 2673 . 2 (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
92, 8syl6reqr 2704 1 (𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cun 3605  c0 3948  {csn 4210  {cpr 4212   × cxp 5141  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1o 7605  df-2o 7606  df-cda 9028
This theorem is referenced by:  pwcda1  9054  unctb  9065  infcdaabs  9066  ackbij1lem5  9084  fin56  9253
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