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Theorem djuassen 10219
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 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵𝐶)))

Proof of Theorem djuassen
StepHypRef Expression
1 0ex 5307 . . . . . 6 ∅ ∈ V
2 simp1 1137 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
3 xpsnen2g 9105 . . . . . 6 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ≈ 𝐴)
41, 2, 3sylancr 587 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐴) ≈ 𝐴)
54ensymd 9045 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6 1oex 8516 . . . . . . 7 1o ∈ V
7 snex 5436 . . . . . . . 8 {∅} ∈ V
8 simp2 1138 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
9 xpexg 7770 . . . . . . . 8 (({∅} ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ∈ V)
107, 8, 9sylancr 587 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ∈ V)
11 xpsnen2g 9105 . . . . . . 7 ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
126, 10, 11sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13 xpsnen2g 9105 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
141, 8, 13sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15 entr 9046 . . . . . 6 ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1612, 14, 15syl2anc 584 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1716ensymd 9045 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18 xp01disjl 8530 . . . . 5 (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
1918a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20 djuenun 10211 . . . 4 ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
215, 17, 19, 20syl3anc 1373 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22 snex 5436 . . . . . . 7 {1o} ∈ V
23 simp3 1139 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
24 xpexg 7770 . . . . . . 7 (({1o} ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ∈ V)
2522, 23, 24sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ∈ V)
26 xpsnen2g 9105 . . . . . 6 ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
276, 25, 26sylancr 587 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28 xpsnen2g 9105 . . . . . 6 ((1o ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
296, 23, 28sylancr 587 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30 entr 9046 . . . . 5 ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3127, 29, 30syl2anc 584 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3231ensymd 9045 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33 indir 4286 . . . . 5 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34 xp01disjl 8530 . . . . . . 7 (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35 xp01disjl 8530 . . . . . . . . 9 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
3635xpeq2i 5712 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37 xpindi 5844 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38 xp0 6178 . . . . . . . 8 ({1o} × ∅) = ∅
3936, 37, 383eqtr3i 2773 . . . . . . 7 (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4034, 39uneq12i 4166 . . . . . 6 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41 un0 4394 . . . . . 6 (∅ ∪ ∅) = ∅
4240, 41eqtri 2765 . . . . 5 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
4333, 42eqtri 2765 . . . 4 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4443a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅)
45 djuenun 10211 . . 3 (((𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∧ 𝐶 ≈ ({1o} × ({1o} × 𝐶)) ∧ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅) → ((𝐴𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
4621, 32, 44, 45syl3anc 1373 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47 df-dju 9941 . . . . . 6 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
4847xpeq2i 5712 . . . . 5 ({1o} × (𝐵𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49 xpundi 5754 . . . . 5 ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5048, 49eqtri 2765 . . . 4 ({1o} × (𝐵𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5150uneq2i 4165 . . 3 (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52 df-dju 9941 . . 3 (𝐴 ⊔ (𝐵𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶)))
53 unass 4172 . . 3 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
5451, 52, 533eqtr4i 2775 . 2 (𝐴 ⊔ (𝐵𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
5546, 54breqtrrdi 5185 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  cin 3950  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  1oc1o 8499  cen 8982  cdju 9938
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-en 8986  df-dju 9941
This theorem is referenced by:  nnadju  10238
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