Theorem djuassen 7329

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 4171	. . . . . 6 ⊢ ∅ ∈ V
2		simp1 1000	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
3		xpsnen2g 6924	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴)
4	1, 2, 3	sylancr 414	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐴) ≈ 𝐴)
5	4	ensymd 6875	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6		1oex 6510	. . . . . . 7 ⊢ 1o ∈ V
7	1	snex 4229	. . . . . . . 8 ⊢ {∅} ∈ V
8		simp2 1001	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
9		xpexg 4789	. . . . . . . 8 ⊢ (({∅} ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ∈ V)
10	7, 8, 9	sylancr 414	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ∈ V)
11		xpsnen2g 6924	. . . . . . 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 6924	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
14	1, 8, 13	sylancr 414	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15		entr 6876	. . . . . 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 6875	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18		xp01disjl 6520	. . . . 5 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
19	18	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20		djuenun 7324	. . . 4 ⊢ ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
21	5, 17, 19, 20	syl3anc 1250	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ⊔ 𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22	6	snex 4229	. . . . . . 7 ⊢ {1o} ∈ V
23		simp3 1002	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
24		xpexg 4789	. . . . . . 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 6924	. . . . . 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 6924	. . . . . 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 6876	. . . . 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 6875	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33		indir 3422	. . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34		xp01disjl 6520	. . . . . . 7 ⊢ (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35		xp01disjl 6520	. . . . . . . . 9 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
36	35	xpeq2i 4696	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37		xpindi 4813	. . . . . . . 8 ⊢ ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38		xp0 5102	. . . . . . . 8 ⊢ ({1o} × ∅) = ∅
39	36, 37, 38	3eqtr3i 2234	. . . . . . 7 ⊢ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
40	34, 39	uneq12i 3325	. . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41		un0 3494	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
42	40, 41	eqtri 2226	. . . . 5 ⊢ ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
43	33, 42	eqtri 2226	. . . 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 7324	. . 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 1250	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47		df-dju 7140	. . . . . 6 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
48	47	xpeq2i 4696	. . . . 5 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49		xpundi 4731	. . . . 5 ⊢ ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
50	48, 49	eqtri 2226	. . . 4 ⊢ ({1o} × (𝐵 ⊔ 𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
51	50	uneq2i 3324	. . 3 ⊢ (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52		df-dju 7140	. . 3 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵 ⊔ 𝐶)))
53		unass 3330	. . 3 ⊢ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
54	51, 52, 53	3eqtr4i 2236	. 2 ⊢ (𝐴 ⊔ (𝐵 ⊔ 𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
55	46, 54	breqtrrdi 4086	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ∪ cun 3164   ∩ cin 3165  ∅c0 3460  {csn 3633   class class class wbr 4044   × cxp 4673  1_oc1o 6495   ≈ cen 6825   ⊔ cdju 7139

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-er 6620  df-en 6828  df-dju 7140  df-inl 7149  df-inr 7150

This theorem is referenced by: (None)