Theorem xp2cda 9002

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 8992	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2	1	anidms 677	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
3		df2o3 7573	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
4		df-pr 4180	. . . . 5 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
5	3, 4	eqtri 2644	. . . 4 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
6	5	xpeq2i 5136	. . 3 ⊢ (𝐴 × 2𝑜) = (𝐴 × ({∅} ∪ {1𝑜}))
7		xpundi 5171	. . 3 ⊢ (𝐴 × ({∅} ∪ {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
8	6, 7	eqtri 2644	. 2 ⊢ (𝐴 × 2𝑜) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))
9	2, 8	syl6reqr 2675	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ∪ cun 3572  ∅c0 3915  {csn 4177  {cpr 4179   × cxp 5112  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-cda 8990

This theorem is referenced by:  pwcda1  9016  unctb  9027  infcdaabs  9028  ackbij1lem5  9046  fin56  9215