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Theorem cdaassen 8955
 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 1059 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2 0ex 4755 . . . . . 6 ∅ ∈ V
3 xpsneng 7996 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
41, 2, 3sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × {∅}) ≈ 𝐴)
54ensymd 7958 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ≈ (𝐴 × {∅}))
6 simp2 1060 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
7 snex 4874 . . . . . . . 8 {∅} ∈ V
8 xpexg 6920 . . . . . . . 8 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
96, 7, 8sylancl 693 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
10 1on 7519 . . . . . . 7 1𝑜 ∈ On
11 xpsneng 7996 . . . . . . 7 (((𝐵 × {∅}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
129, 10, 11sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}))
13 xpsneng 7996 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
146, 2, 13sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
15 entr 7959 . . . . . 6 ((((𝐵 × {∅}) × {1𝑜}) ≈ (𝐵 × {∅}) ∧ (𝐵 × {∅}) ≈ 𝐵) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1612, 14, 15syl2anc 692 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) × {1𝑜}) ≈ 𝐵)
1716ensymd 7958 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}))
18 xp01disj 7528 . . . . 5 ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅
1918a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅)
20 cdaenun 8947 . . . 4 ((𝐴 ≈ (𝐴 × {∅}) ∧ 𝐵 ≈ ((𝐵 × {∅}) × {1𝑜}) ∧ ((𝐴 × {∅}) ∩ ((𝐵 × {∅}) × {1𝑜})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
215, 17, 19, 20syl3anc 1323 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})))
22 simp3 1061 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
23 snex 4874 . . . . . . 7 {1𝑜} ∈ V
24 xpexg 6920 . . . . . . 7 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
2522, 23, 24sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
26 xpsneng 7996 . . . . . 6 (((𝐶 × {1𝑜}) ∈ V ∧ 1𝑜 ∈ On) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
2725, 10, 26sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}))
28 xpsneng 7996 . . . . . 6 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
2922, 10, 28sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
30 entr 7959 . . . . 5 ((((𝐶 × {1𝑜}) × {1𝑜}) ≈ (𝐶 × {1𝑜}) ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3127, 29, 30syl2anc 692 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐶 × {1𝑜}) × {1𝑜}) ≈ 𝐶)
3231ensymd 7958 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ((𝐶 × {1𝑜}) × {1𝑜}))
33 indir 3856 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})))
34 xp01disj 7528 . . . . . 6 ((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
35 xp01disj 7528 . . . . . . . 8 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
3635xpeq1i 5100 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (∅ × {1𝑜})
37 xpindir 5221 . . . . . . 7 (((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))
38 0xp 5165 . . . . . . 7 (∅ × {1𝑜}) = ∅
3936, 37, 383eqtr3i 2651 . . . . . 6 (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4034, 39uneq12i 3748 . . . . 5 (((𝐴 × {∅}) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) ∪ (((𝐵 × {∅}) × {1𝑜}) ∩ ((𝐶 × {1𝑜}) × {1𝑜}))) = (∅ ∪ ∅)
41 un0 3944 . . . . 5 (∅ ∪ ∅) = ∅
4233, 40, 413eqtri 2647 . . . 4 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅
4342a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∩ ((𝐶 × {1𝑜}) × {1𝑜})) = ∅)
44 cdaenun 8947 . . 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 1323 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
46 ovex 6638 . . . . 5 (𝐵 +𝑐 𝐶) ∈ V
47 cdaval 8943 . . . . 5 ((𝐴𝑉 ∧ (𝐵 +𝑐 𝐶) ∈ V) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
4846, 47mpan2 706 . . . 4 (𝐴𝑉 → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})))
49 cdaval 8943 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
5049xpeq1d 5103 . . . . . . 7 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}))
51 xpundir 5138 . . . . . . 7 (((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))
5250, 51syl6eq 2671 . . . . . 6 ((𝐵𝑊𝐶𝑋) → ((𝐵 +𝑐 𝐶) × {1𝑜}) = (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5352uneq2d 3750 . . . . 5 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜}))))
54 unass 3753 . . . . 5 (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})) = ((𝐴 × {∅}) ∪ (((𝐵 × {∅}) × {1𝑜}) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5553, 54syl6eqr 2673 . . . 4 ((𝐵𝑊𝐶𝑋) → ((𝐴 × {∅}) ∪ ((𝐵 +𝑐 𝐶) × {1𝑜})) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5648, 55sylan9eq 2675 . . 3 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
57563impb 1257 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 +𝑐 (𝐵 +𝑐 𝐶)) = (((𝐴 × {∅}) ∪ ((𝐵 × {∅}) × {1𝑜})) ∪ ((𝐶 × {1𝑜}) × {1𝑜})))
5845, 57breqtrrd 4646 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  Vcvv 3189   ∪ cun 3557   ∩ cin 3558  ∅c0 3896  {csn 4153   class class class wbr 4618   × cxp 5077  Oncon0 5687  (class class class)co 6610  1𝑜c1o 7505   ≈ cen 7903   +𝑐 ccda 8940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1o 7512  df-er 7694  df-en 7907  df-cda 8941 This theorem is referenced by: (None)
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