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Theorem cdaassen 8887
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 1054 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2 0ex 4718 . . . . . 6 ∅ ∈ V
3 xpsneng 7930 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
41, 2, 3sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × {∅}) ≈ 𝐴)
54ensymd 7893 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ (𝐴 × {∅}))
6 simp2 1055 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
7 snex 4835 . . . . . . . 8 {∅} ∈ V
8 xpexg 6858 . . . . . . . 8 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
96, 7, 8sylancl 693 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
10 1on 7454 . . . . . . 7 1𝑜 ∈ On
11 xpsneng 7930 . . . . . . 7 (((𝐵 × {∅}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
129, 10, 11sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
13 xpsneng 7930 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
146, 2, 13sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
15 entr 7894 . . . . . 6 ((((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}) ∧ (𝐵 × {∅}) ≈ 𝐵) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1612, 14, 15syl2anc 691 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1716ensymd 7893 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}))
18 xp01disj 7463 . . . . 5 ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅
1918a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅)
20 cdaenun 8879 . . . 4 ((𝐴 ≈ (𝐴 × {∅}) ∧ 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}) ∧ ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
215, 17, 19, 20syl3anc 1318 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
22 simp3 1056 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
23 snex 4835 . . . . . . 7 {1𝑜} ∈ V
24 xpexg 6858 . . . . . . 7 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
2522, 23, 24sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
26 xpsneng 7930 . . . . . 6 (((𝐶 × {1𝑜}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
2725, 10, 26sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
28 xpsneng 7930 . . . . . 6 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
2922, 10, 28sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
30 entr 7894 . . . . 5 ((((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}) ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3127, 29, 30syl2anc 691 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3231ensymd 7893 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}))
33 indir 3834 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})))
34 xp01disj 7463 . . . . . 6 ((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
35 xp01disj 7463 . . . . . . . 8 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
3635xpeq1i 5059 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (∅ × {1𝑜})
37 xpindir 5178 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))
38 0xp 5122 . . . . . . 7 (∅ × {1𝑜}) = ∅
3936, 37, 383eqtr3i 2640 . . . . . 6 (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4034, 39uneq12i 3727 . . . . 5 (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))) = (∅ ∪ ∅)
41 un0 3919 . . . . 5 (∅ ∪ ∅) = ∅
4233, 40, 413eqtri 2636 . . . 4 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4342a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅)
44 cdaenun 8879 . . 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 1318 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
46 ovex 6577 . . . . 5 (𝐵 +𝑐 𝐶) ∈ V
47 cdaval 8875 . . . . 5 ((𝐴𝑉 ∧ (𝐵 +𝑐 𝐶) ∈ V) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
4846, 47mpan2 703 . . . 4 (𝐴𝑉 → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
49 cdaval 8875 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
5049xpeq1d 5062 . . . . . . 7 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}))
51 xpundir 5095 . . . . . . 7 (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))
5250, 51syl6eq 2660 . . . . . 6 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5352uneq2d 3729 . . . . 5 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))))
54 unass 3732 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5553, 54syl6eqr 2662 . . . 4 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5648, 55sylan9eq 2664 . . 3 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
57563impb 1252 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5845, 57breqtrrd 4611 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 1031   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  cin 3539  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440  cen 7838   +𝑐 ccda 8872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-er 7629  df-en 7842  df-cda 8873
This theorem is referenced by: (None)
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