Theorem djuassen 7422

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 4214	. . . . . 6 ⊢ ∅ ∈ V
2		simp1 1021	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
3		xpsnen2g 7008	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴)
4	1, 2, 3	sylancr 414	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐴) ≈ 𝐴)
5	4	ensymd 6952	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6		1oex 6585	. . . . . . 7 ⊢ 1o ∈ V
7	1	snex 4273	. . . . . . . 8 ⊢ {∅} ∈ V
8		simp2 1022	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
9		xpexg 4838	. . . . . . . 8 ⊢ (({∅} ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ∈ V)
10	7, 8, 9	sylancr 414	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ∈ V)
11		xpsnen2g 7008	. . . . . . 7 ⊢ ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
12	6, 10, 11	sylancr 414	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13		xpsnen2g 7008	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
14	1, 8, 13	sylancr 414	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15		entr 6953	. . . . . 6 ⊢ ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
16	12, 14, 15	syl2anc 411	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
17	16	ensymd 6952	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18		xp01disjl 6597	. . . . 5 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
19	18	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20		djuenun 7417	. . . 4 ⊢ ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
21	5, 17, 19, 20	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22	6	snex 4273	. . . . . . 7 ⊢ {1o} ∈ V
23		simp3 1023	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
24		xpexg 4838	. . . . . . 7 ⊢ (({1o} ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
25	22, 23, 24	sylancr 414	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
26		xpsnen2g 7008	. . . . . 6 ⊢ ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
27	6, 25, 26	sylancr 414	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28		xpsnen2g 7008	. . . . . 6 ⊢ ((1o ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
29	6, 23, 28	sylancr 414	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30		entr 6953	. . . . 5 ⊢ ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
31	27, 29, 30	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
32	31	ensymd 6952	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33		indir 3454	. . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34		xp01disjl 6597	. . . . . . 7 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35		xp01disjl 6597	. . . . . . . . 9 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
36	35	xpeq2i 4744	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37		xpindi 4863	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38		xp0 5154	. . . . . . . 8 ⊢ ({1o} × ∅) = ∅
39	36, 37, 38	3eqtr3i 2258	. . . . . . 7 ⊢ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
40	34, 39	uneq12i 3357	. . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41		un0 3526	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
42	40, 41	eqtri 2250	. . . . 5 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
43	33, 42	eqtri 2250	. . . 4 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅
44	43	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅)
45		djuenun 7417	. . 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 1271	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47		df-dju 7228	. . . . . 6 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
48	47	xpeq2i 4744	. . . . 5 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49		xpundi 4780	. . . . 5 ⊢ ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
50	48, 49	eqtri 2250	. . . 4 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
51	50	uneq2i 3356	. . 3 ⊢ (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52		df-dju 7228	. . 3 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶)))
53		unass 3362	. . 3 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
54	51, 52, 53	3eqtr4i 2260	. 2 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
55	46, 54	breqtrrdi 4128	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196   ∩ cin 3197  ∅c0 3492  {csn 3667   class class class wbr 4086   × cxp 4721  1_oc1o 6570   ≈ cen 6902   ⊔ cdju 7227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-1o 6577  df-er 6697  df-en 6905  df-dju 7228  df-inl 7237  df-inr 7238

This theorem is referenced by: (None)