Theorem djuassen 9604

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
djuassen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))

Proof of Theorem djuassen

Step	Hyp	Ref	Expression
1		0ex 5211	. . . . . 6 ⊢ ∅ ∈ V
2		simp1 1132	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
3		xpsnen2g 8610	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴)
4	1, 2, 3	sylancr 589	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐴) ≈ 𝐴)
5	4	ensymd 8560	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6		1oex 8110	. . . . . . 7 ⊢ 1o ∈ V
7		snex 5332	. . . . . . . 8 ⊢ {∅} ∈ V
8		simp2 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
9		xpexg 7473	. . . . . . . 8 ⊢ (({∅} ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ∈ V)
10	7, 8, 9	sylancr 589	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ∈ V)
11		xpsnen2g 8610	. . . . . . 7 ⊢ ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
12	6, 10, 11	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13		xpsnen2g 8610	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
14	1, 8, 13	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15		entr 8561	. . . . . 6 ⊢ ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
16	12, 14, 15	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
17	16	ensymd 8560	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18		xp01disjl 8121	. . . . 5 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
19	18	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20		djuenun 9596	. . . 4 ⊢ ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
21	5, 17, 19, 20	syl3anc 1367	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22		snex 5332	. . . . . . 7 ⊢ {1o} ∈ V
23		simp3 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
24		xpexg 7473	. . . . . . 7 ⊢ (({1o} ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
25	22, 23, 24	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
26		xpsnen2g 8610	. . . . . 6 ⊢ ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
27	6, 25, 26	sylancr 589	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28		xpsnen2g 8610	. . . . . 6 ⊢ ((1o ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
29	6, 23, 28	sylancr 589	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30		entr 8561	. . . . 5 ⊢ ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
31	27, 29, 30	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
32	31	ensymd 8560	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33		indir 4252	. . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34		xp01disjl 8121	. . . . . . 7 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35		xp01disjl 8121	. . . . . . . . 9 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
36	35	xpeq2i 5582	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37		xpindi 5704	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38		xp0 6015	. . . . . . . 8 ⊢ ({1o} × ∅) = ∅
39	36, 37, 38	3eqtr3i 2852	. . . . . . 7 ⊢ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
40	34, 39	uneq12i 4137	. . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41		un0 4344	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
42	40, 41	eqtri 2844	. . . . 5 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
43	33, 42	eqtri 2844	. . . 4 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅
44	43	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅)
45		djuenun 9596	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∧ 𝐶 ≈ ({1o} × ({1o} × 𝐶)) ∧ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
46	21, 32, 44, 45	syl3anc 1367	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47		df-dju 9330	. . . . . 6 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
48	47	xpeq2i 5582	. . . . 5 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49		xpundi 5620	. . . . 5 ⊢ ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
50	48, 49	eqtri 2844	. . . 4 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
51	50	uneq2i 4136	. . 3 ⊢ (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52		df-dju 9330	. . 3 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶)))
53		unass 4142	. . 3 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
54	51, 52, 53	3eqtr4i 2854	. 2 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
55	46, 54	breqtrrdi 5108	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  {csn 4567   class class class wbr 5066   × cxp 5553  1_oc1o 8095   ≈ cen 8506   ⊔ cdju 9327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-1st 7689  df-2nd 7690  df-1o 8102  df-er 8289  df-en 8510  df-dju 9330

This theorem is referenced by: (None)