Theorem cdaassen 9004

Description: Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdaassen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))

Proof of Theorem cdaassen

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
2		0ex 4790	. . . . . 6 ⊢ ∅ ∈ V
3		xpsneng 8045	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
4	1, 2, 3	sylancl 694	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × {∅}) ≈ 𝐴)
5	4	ensymd 8007	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ (𝐴 × {∅}))
6		simp2 1062	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
7		snex 4908	. . . . . . . 8 ⊢ {∅} ∈ V
8		xpexg 6960	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
9	6, 7, 8	sylancl 694	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × {∅}) ∈ V)
10		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
11		xpsneng 8045	. . . . . . 7 ⊢ (((𝐵 × {∅}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
12	9, 10, 11	sylancl 694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
13		xpsneng 8045	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
14	6, 2, 13	sylancl 694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × {∅}) ≈ 𝐵)
15		entr 8008	. . . . . 6 ⊢ ((((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}) ∧ (𝐵 × {∅}) ≈ 𝐵) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
16	12, 14, 15	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
17	16	ensymd 8007	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}))
18		xp01disj 7576	. . . . 5 ⊢ ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅
19	18	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅)
20		cdaenun 8996	. . . 4 ⊢ ((𝐴 ≈ (𝐴 × {∅}) ∧ 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}) ∧ ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
21	5, 17, 19, 20	syl3anc 1326	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
22		simp3 1063	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
23		snex 4908	. . . . . . 7 ⊢ {1𝑜} ∈ V
24		xpexg 6960	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
25	22, 23, 24	sylancl 694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐶 × {1𝑜}) ∈ V)
26		xpsneng 8045	. . . . . 6 ⊢ (((𝐶 × {1𝑜}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
27	25, 10, 26	sylancl 694	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
28		xpsneng 8045	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
29	22, 10, 28	sylancl 694	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
30		entr 8008	. . . . 5 ⊢ ((((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}) ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
31	27, 29, 30	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
32	31	ensymd 8007	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}))
33		indir 3875	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})))
34		xp01disj 7576	. . . . . 6 ⊢ ((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
35		xp01disj 7576	. . . . . . . 8 ⊢ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
36	35	xpeq1i 5135	. . . . . . 7 ⊢ (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (∅ × {1𝑜})
37		xpindir 5256	. . . . . . 7 ⊢ (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))
38		0xp 5199	. . . . . . 7 ⊢ (∅ × {1𝑜}) = ∅
39	36, 37, 38	3eqtr3i 2652	. . . . . 6 ⊢ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
40	34, 39	uneq12i 3765	. . . . 5 ⊢ (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))) = (∅ ∪ ∅)
41		un0 3967	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
42	33, 40, 41	3eqtri 2648	. . . 4 ⊢ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
43	42	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅)
44		cdaenun 8996	. . 3 ⊢ (((𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∧ 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}) ∧ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
45	21, 32, 43, 44	syl3anc 1326	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
46		ovex 6678	. . . . 5 ⊢ (𝐵 +𝑐 𝐶) ∈ V
47		cdaval 8992	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 +𝑐 𝐶) ∈ V) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
48	46, 47	mpan2 707	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
49		cdaval 8992	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
50	49	xpeq1d 5138	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}))
51		xpundir 5172	. . . . . . 7 ⊢ (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))
52	50, 51	syl6eq 2672	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
53	52	uneq2d 3767	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))))
54		unass 3770	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
55	53, 54	syl6eqr 2674	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
56	48, 55	sylan9eq 2676	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
57	56	3impb 1260	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
58	45, 57	breqtrrd 4681	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   +_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-3or 1038  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-ne 2795  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by: (None)