MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djuassen Structured version   Visualization version   GIF version

Theorem djuassen 10248
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 5325 . . . . . 6 ∅ ∈ V
2 simp1 1136 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
3 xpsnen2g 9131 . . . . . 6 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} × 𝐴) ≈ 𝐴)
41, 2, 3sylancr 586 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐴) ≈ 𝐴)
54ensymd 9065 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ ({∅} × 𝐴))
6 1oex 8532 . . . . . . 7 1o ∈ V
7 snex 5451 . . . . . . . 8 {∅} ∈ V
8 simp2 1137 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
9 xpexg 7785 . . . . . . . 8 (({∅} ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ∈ V)
107, 8, 9sylancr 586 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ∈ V)
11 xpsnen2g 9131 . . . . . . 7 ((1o ∈ V ∧ ({∅} × 𝐵) ∈ V) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
126, 10, 11sylancr 586 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵))
13 xpsnen2g 9131 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
141, 8, 13sylancr 586 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
15 entr 9066 . . . . . 6 ((({1o} × ({∅} × 𝐵)) ≈ ({∅} × 𝐵) ∧ ({∅} × 𝐵) ≈ 𝐵) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1612, 14, 15syl2anc 583 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({∅} × 𝐵)) ≈ 𝐵)
1716ensymd 9065 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({1o} × ({∅} × 𝐵)))
18 xp01disjl 8548 . . . . 5 (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅
1918a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅)
20 djuenun 10240 . . . 4 ((𝐴 ≈ ({∅} × 𝐴) ∧ 𝐵 ≈ ({1o} × ({∅} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × ({∅} × 𝐵))) = ∅) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
215, 17, 19, 20syl3anc 1371 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝐵) ≈ (({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))))
22 snex 5451 . . . . . . 7 {1o} ∈ V
23 simp3 1138 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
24 xpexg 7785 . . . . . . 7 (({1o} ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ∈ V)
2522, 23, 24sylancr 586 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ∈ V)
26 xpsnen2g 9131 . . . . . 6 ((1o ∈ V ∧ ({1o} × 𝐶) ∈ V) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
276, 25, 26sylancr 586 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶))
28 xpsnen2g 9131 . . . . . 6 ((1o ∈ V ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
296, 23, 28sylancr 586 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
30 entr 9066 . . . . 5 ((({1o} × ({1o} × 𝐶)) ≈ ({1o} × 𝐶) ∧ ({1o} × 𝐶) ≈ 𝐶) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3127, 29, 30syl2anc 583 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × ({1o} × 𝐶)) ≈ 𝐶)
3231ensymd 9065 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × ({1o} × 𝐶)))
33 indir 4305 . . . . 5 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))))
34 xp01disjl 8548 . . . . . . 7 (({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) = ∅
35 xp01disjl 8548 . . . . . . . . 9 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
3635xpeq2i 5727 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ({1o} × ∅)
37 xpindi 5858 . . . . . . . 8 ({1o} × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))
38 xp0 6189 . . . . . . . 8 ({1o} × ∅) = ∅
3936, 37, 383eqtr3i 2776 . . . . . . 7 (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4034, 39uneq12i 4189 . . . . . 6 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = (∅ ∪ ∅)
41 un0 4417 . . . . . 6 (∅ ∪ ∅) = ∅
4240, 41eqtri 2768 . . . . 5 ((({∅} × 𝐴) ∩ ({1o} × ({1o} × 𝐶))) ∪ (({1o} × ({∅} × 𝐵)) ∩ ({1o} × ({1o} × 𝐶)))) = ∅
4333, 42eqtri 2768 . . . 4 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅
4443a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∩ ({1o} × ({1o} × 𝐶))) = ∅)
45 djuenun 10240 . . 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 1371 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))))
47 df-dju 9970 . . . . . 6 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
4847xpeq2i 5727 . . . . 5 ({1o} × (𝐵𝐶)) = ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
49 xpundi 5768 . . . . 5 ({1o} × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5048, 49eqtri 2768 . . . 4 ({1o} × (𝐵𝐶)) = (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶)))
5150uneq2i 4188 . . 3 (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
52 df-dju 9970 . . 3 (𝐴 ⊔ (𝐵𝐶)) = (({∅} × 𝐴) ∪ ({1o} × (𝐵𝐶)))
53 unass 4195 . . 3 ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶))) = (({∅} × 𝐴) ∪ (({1o} × ({∅} × 𝐵)) ∪ ({1o} × ({1o} × 𝐶))))
5451, 52, 533eqtr4i 2778 . 2 (𝐴 ⊔ (𝐵𝐶)) = ((({∅} × 𝐴) ∪ ({1o} × ({∅} × 𝐵))) ∪ ({1o} × ({1o} × 𝐶)))
5546, 54breqtrrdi 5208 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1537  wcel 2108  Vcvv 3488  cun 3974  cin 3975  c0 4352  {csn 4648   class class class wbr 5166   × cxp 5698  1oc1o 8515  cen 9000  cdju 9967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031  df-1o 8522  df-er 8763  df-en 9004  df-dju 9970
This theorem is referenced by:  nnadju  10267
  Copyright terms: Public domain W3C validator