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Theorem cdacomen 8886
 Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdacomen (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)

Proof of Theorem cdacomen
StepHypRef Expression
1 1on 7454 . . . . 5 1𝑜 ∈ On
2 xpsneng 7930 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
31, 2mpan2 703 . . . 4 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
4 0ex 4718 . . . . 5 ∅ ∈ V
5 xpsneng 7930 . . . . 5 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
64, 5mpan2 703 . . . 4 (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7 ensym 7891 . . . . 5 ((𝐴 × {1𝑜}) ≈ 𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 ensym 7891 . . . . 5 ((𝐵 × {∅}) ≈ 𝐵𝐵 ≈ (𝐵 × {∅}))
9 incom 3767 . . . . . . 7 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜}))
10 xp01disj 7463 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅
119, 10eqtri 2632 . . . . . 6 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅
12 cdaenun 8879 . . . . . 6 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
1311, 12mp3an3 1405 . . . . 5 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
147, 8, 13syl2an 493 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
153, 6, 14syl2an 493 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
16 cdaval 8875 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
1716ancoms 468 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
18 uncom 3719 . . . 4 ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))
1917, 18syl6eq 2660 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
2015, 19breqtrrd 4611 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
214enref 7874 . . . 4 ∅ ≈ ∅
2221a1i 11 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23 cdafn 8874 . . . . 5 +𝑐 Fn (V × V)
24 fndm 5904 . . . . 5 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2523, 24ax-mp 5 . . . 4 dom +𝑐 = (V × V)
2625ndmov 6716 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27 ancom 465 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
2825ndmov 6716 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
2927, 28sylnbi 319 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
3022, 26, 293brtr4d 4615 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
3120, 30pm2.61i 175 1 (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  dom cdm 5038  Oncon0 5640   Fn wfn 5799  (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-csb 3500  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-iun 4457  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-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-cda 8873 This theorem is referenced by:  cdadom2  8892  cdalepw  8901  infcda  8913  alephadd  9278  gchdomtri  9330  pwxpndom  9367  gchpwdom  9371  gchhar  9380
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