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Theorem xp2cda 8862
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 8852 . . 3 ((𝐴𝑉𝐴𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
21anidms 674 . 2 (𝐴𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3 df2o3 7437 . . . . 5 2𝑜 = {∅, 1𝑜}
4 df-pr 4127 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
53, 4eqtri 2631 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
65xpeq2i 5050 . . 3 (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7 xpundi 5084 . . 3 (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
86, 7eqtri 2631 . 2 (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
92, 8syl6reqr 2662 1 (𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cun 3537  c0 3873  {csn 4124  {cpr 4126   × cxp 5026  (class class class)co 6527  1𝑜c1o 7417  2𝑜c2o 7418   +𝑐 ccda 8849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-suc 5632  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1o 7424  df-2o 7425  df-cda 8850
This theorem is referenced by:  pwcda1  8876  unctb  8887  infcdaabs  8888  ackbij1lem5  8906  fin56  9075
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