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Theorem cdaassen 9042
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
cdaassen ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))

Proof of Theorem cdaassen
StepHypRef Expression
1 simp1 1081 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2 0ex 4823 . . . . . 6 ∅ ∈ V
3 xpsneng 8086 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
41, 2, 3sylancl 695 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × {∅}) ≈ 𝐴)
54ensymd 8048 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ (𝐴 × {∅}))
6 simp2 1082 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
7 snex 4938 . . . . . . . 8 {∅} ∈ V
8 xpexg 7002 . . . . . . . 8 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
96, 7, 8sylancl 695 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
10 1on 7612 . . . . . . 7 1𝑜 ∈ On
11 xpsneng 8086 . . . . . . 7 (((𝐵 × {∅}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
129, 10, 11sylancl 695 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
13 xpsneng 8086 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
146, 2, 13sylancl 695 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
15 entr 8049 . . . . . 6 ((((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}) ∧ (𝐵 × {∅}) ≈ 𝐵) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1612, 14, 15syl2anc 694 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1716ensymd 8048 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}))
18 xp01disj 7621 . . . . 5 ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅
1918a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅)
20 cdaenun 9034 . . . 4 ((𝐴 ≈ (𝐴 × {∅}) ∧ 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}) ∧ ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
215, 17, 19, 20syl3anc 1366 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
22 simp3 1083 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
23 snex 4938 . . . . . . 7 {1𝑜} ∈ V
24 xpexg 7002 . . . . . . 7 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
2522, 23, 24sylancl 695 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
26 xpsneng 8086 . . . . . 6 (((𝐶 × {1𝑜}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
2725, 10, 26sylancl 695 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
28 xpsneng 8086 . . . . . 6 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
2922, 10, 28sylancl 695 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
30 entr 8049 . . . . 5 ((((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}) ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3127, 29, 30syl2anc 694 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3231ensymd 8048 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}))
33 indir 3908 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})))
34 xp01disj 7621 . . . . . 6 ((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
35 xp01disj 7621 . . . . . . . 8 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
3635xpeq1i 5169 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (∅ × {1𝑜})
37 xpindir 5289 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))
38 0xp 5233 . . . . . . 7 (∅ × {1𝑜}) = ∅
3936, 37, 383eqtr3i 2681 . . . . . 6 (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4034, 39uneq12i 3798 . . . . 5 (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))) = (∅ ∪ ∅)
41 un0 4000 . . . . 5 (∅ ∪ ∅) = ∅
4233, 40, 413eqtri 2677 . . . 4 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4342a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅)
44 cdaenun 9034 . . 3 (((𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∧ 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}) ∧ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
4521, 32, 43, 44syl3anc 1366 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
46 ovex 6718 . . . . 5 (𝐵 +𝑐 𝐶) ∈ V
47 cdaval 9030 . . . . 5 ((𝐴𝑉 ∧ (𝐵 +𝑐 𝐶) ∈ V) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
4846, 47mpan2 707 . . . 4 (𝐴𝑉 → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
49 cdaval 9030 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
5049xpeq1d 5172 . . . . . . 7 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}))
51 xpundir 5206 . . . . . . 7 (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))
5250, 51syl6eq 2701 . . . . . 6 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5352uneq2d 3800 . . . . 5 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))))
54 unass 3803 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5553, 54syl6eqr 2703 . . . 4 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5648, 55sylan9eq 2705 . . 3 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
57563impb 1279 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5845, 57breqtrrd 4713 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598  cen 7994   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1o 7605  df-er 7787  df-en 7998  df-cda 9028
This theorem is referenced by: (None)
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