Theorem xp2cda 9186

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 9176	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2	1	anidms 680	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3		df2o3 7734	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
4		df-pr 4316	. . . . 5 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
5	3, 4	eqtri 2774	. . . 4 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
6	5	xpeq2i 5285	. . 3 ⊢ (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7		xpundi 5320	. . 3 ⊢ (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
8	6, 7	eqtri 2774	. 2 ⊢ (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
9	2, 8	syl6reqr 2805	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Syntax hints:   → wi 4   = wceq 1624   ∈ wcel 2131   ∪ cun 3705  ∅c0 4050  {csn 4313  {cpr 4315   × cxp 5256  (class class class)co 6805  1_𝑜c1o 7714  2_𝑜c2o 7715   +_𝑐 ccda 9173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-suc 5882  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1o 7721  df-2o 7722  df-cda 9174

This theorem is referenced by:  pwcda1  9200  unctb  9211  infcdaabs  9212  ackbij1lem5  9230  fin56  9399