Theorem xp2cda 8885

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	⊢ (𝐴 ∈ 𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Proof of Theorem xp2cda

Step	Hyp	Ref	Expression
1		cdaval 8875	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2	1	anidms 675	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3		df2o3 7460	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
4		df-pr 4128	. . . . 5 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
5	3, 4	eqtri 2632	. . . 4 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
6	5	xpeq2i 5060	. . 3 ⊢ (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7		xpundi 5094	. . 3 ⊢ (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
8	6, 7	eqtri 2632	. 2 ⊢ (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
9	2, 8	syl6reqr 2663	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127   × cxp 5036  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-cda 8873

This theorem is referenced by:  pwcda1  8899  unctb  8910  infcdaabs  8911  ackbij1lem5  8929  fin56  9098