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Theorem xp2cda 9287
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 9277 . . 3 ((𝐴𝑉𝐴𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
21anidms 558 . 2 (𝐴𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3 df2o3 7810 . . . . 5 2𝑜 = {∅, 1𝑜}
4 df-pr 4373 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
53, 4eqtri 2828 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
65xpeq2i 5337 . . 3 (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7 xpundi 5371 . . 3 (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
86, 7eqtri 2828 . 2 (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
92, 8syl6reqr 2859 1 (𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  cun 3767  c0 4116  {csn 4370  {cpr 4372   × cxp 5309  (class class class)co 6874  1𝑜c1o 7789  2𝑜c2o 7790   +𝑐 ccda 9274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-suc 5942  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1o 7796  df-2o 7797  df-cda 9275
This theorem is referenced by:  pwcda1  9301  unctb  9312  infcdaabs  9313  ackbij1lem5  9331  fin56  9500
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