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Theorem djuassen 7147
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 4091 . . . . . 6 ∅ ∈ V
2 simp1 982 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
3 xpsnen2g 6771 . . . . . 6 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ≈ 𝐴)
41, 2, 3sylancr 411 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐴) ≈ 𝐴)
54ensymd 6725 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6 1oex 6368 . . . . . . 7 1o ∈ V
71snex 4146 . . . . . . . 8 {∅} ∈ V
8 simp2 983 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
9 xpexg 4699 . . . . . . . 8 (({∅} ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ∈ V)
107, 8, 9sylancr 411 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ∈ V)
11 xpsnen2g 6771 . . . . . . 7 ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
126, 10, 11sylancr 411 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13 xpsnen2g 6771 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
141, 8, 13sylancr 411 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15 entr 6726 . . . . . 6 ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1612, 14, 15syl2anc 409 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1716ensymd 6725 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18 xp01disjl 6378 . . . . 5 (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
1918a1i 9 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20 djuenun 7142 . . . 4 ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
215, 17, 19, 20syl3anc 1220 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
226snex 4146 . . . . . . 7 {1o} ∈ V
23 simp3 984 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
24 xpexg 4699 . . . . . . 7 (({1o} ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ∈ V)
2522, 23, 24sylancr 411 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ∈ V)
26 xpsnen2g 6771 . . . . . 6 ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
276, 25, 26sylancr 411 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28 xpsnen2g 6771 . . . . . 6 ((1o ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
296, 23, 28sylancr 411 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30 entr 6726 . . . . 5 ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3127, 29, 30syl2anc 409 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3231ensymd 6725 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33 indir 3356 . . . . 5 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34 xp01disjl 6378 . . . . . . 7 (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35 xp01disjl 6378 . . . . . . . . 9 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
3635xpeq2i 4606 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37 xpindi 4720 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38 xp0 5004 . . . . . . . 8 ({1o} × ∅) = ∅
3936, 37, 383eqtr3i 2186 . . . . . . 7 (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4034, 39uneq12i 3259 . . . . . 6 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41 un0 3427 . . . . . 6 (∅ ∪ ∅) = ∅
4240, 41eqtri 2178 . . . . 5 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
4333, 42eqtri 2178 . . . 4 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4443a1i 9 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅)
45 djuenun 7142 . . 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 1220 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47 df-dju 6977 . . . . . 6 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
4847xpeq2i 4606 . . . . 5 ({1o} × (𝐵𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49 xpundi 4641 . . . . 5 ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5048, 49eqtri 2178 . . . 4 ({1o} × (𝐵𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5150uneq2i 3258 . . 3 (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52 df-dju 6977 . . 3 (𝐴 ⊔ (𝐵𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶)))
53 unass 3264 . . 3 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
5451, 52, 533eqtr4i 2188 . 2 (𝐴 ⊔ (𝐵𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
5546, 54breqtrrdi 4006 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 963   = wceq 1335  wcel 2128  Vcvv 2712  cun 3100  cin 3101  c0 3394  {csn 3560   class class class wbr 3965   × cxp 4583  1oc1o 6353  cen 6680  cdju 6976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-er 6477  df-en 6683  df-dju 6977  df-inl 6986  df-inr 6987
This theorem is referenced by: (None)
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