Theorem djuassen 10073

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 5246	. . . . . 6 ⊢ ∅ ∈ V
2		simp1 1136	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
3		xpsnen2g 8987	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴)
4	1, 2, 3	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐴) ≈ 𝐴)
5	4	ensymd 8930	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6		1oex 8398	. . . . . . 7 ⊢ 1o ∈ V
7		snex 5375	. . . . . . . 8 ⊢ {∅} ∈ V
8		simp2 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
9		xpexg 7686	. . . . . . . 8 ⊢ (({∅} ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ∈ V)
10	7, 8, 9	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ∈ V)
11		xpsnen2g 8987	. . . . . . 7 ⊢ ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
12	6, 10, 11	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13		xpsnen2g 8987	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
14	1, 8, 13	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15		entr 8931	. . . . . 6 ⊢ ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
16	12, 14, 15	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
17	16	ensymd 8930	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18		xp01disjl 8410	. . . . 5 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
19	18	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20		djuenun 10065	. . . 4 ⊢ ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
21	5, 17, 19, 20	syl3anc 1373	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22		snex 5375	. . . . . . 7 ⊢ {1o} ∈ V
23		simp3 1138	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
24		xpexg 7686	. . . . . . 7 ⊢ (({1o} ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
25	22, 23, 24	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ∈ V)
26		xpsnen2g 8987	. . . . . 6 ⊢ ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
27	6, 25, 26	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28		xpsnen2g 8987	. . . . . 6 ⊢ ((1o ∈ V ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
29	6, 23, 28	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30		entr 8931	. . . . 5 ⊢ ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
31	27, 29, 30	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
32	31	ensymd 8930	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33		indir 4237	. . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34		xp01disjl 8410	. . . . . . 7 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35		xp01disjl 8410	. . . . . . . . 9 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
36	35	xpeq2i 5646	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37		xpindi 5776	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38		xp0 6107	. . . . . . . 8 ⊢ ({1o} × ∅) = ∅
39	36, 37, 38	3eqtr3i 2760	. . . . . . 7 ⊢ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
40	34, 39	uneq12i 4117	. . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41		un0 4345	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
42	40, 41	eqtri 2752	. . . . 5 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
43	33, 42	eqtri 2752	. . . 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 10065	. . 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 1373	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47		df-dju 9797	. . . . . 6 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
48	47	xpeq2i 5646	. . . . 5 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49		xpundi 5688	. . . . 5 ⊢ ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
50	48, 49	eqtri 2752	. . . 4 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
51	50	uneq2i 4116	. . 3 ⊢ (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52		df-dju 9797	. . 3 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶)))
53		unass 4123	. . 3 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
54	51, 52, 53	3eqtr4i 2762	. 2 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
55	46, 54	breqtrrdi 5134	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901   ∩ cin 3902  ∅c0 4284  {csn 4577   class class class wbr 5092   × cxp 5617  1_oc1o 8381   ≈ cen 8869   ⊔ cdju 9794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-1st 7924  df-2nd 7925  df-1o 8388  df-er 8625  df-en 8873  df-dju 9797

This theorem is referenced by:  nnadju  10092