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Theorem cdacomen 8963
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 7527 . . . . 5 1𝑜 ∈ On
2 xpsneng 8005 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
31, 2mpan2 706 . . . 4 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
4 0ex 4760 . . . . 5 ∅ ∈ V
5 xpsneng 8005 . . . . 5 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
64, 5mpan2 706 . . . 4 (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7 ensym 7965 . . . . 5 ((𝐴 × {1𝑜}) ≈ 𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 ensym 7965 . . . . 5 ((𝐵 × {∅}) ≈ 𝐵𝐵 ≈ (𝐵 × {∅}))
9 incom 3789 . . . . . . 7 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜}))
10 xp01disj 7536 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅
119, 10eqtri 2643 . . . . . 6 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅
12 cdaenun 8956 . . . . . 6 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
1311, 12mp3an3 1410 . . . . 5 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
147, 8, 13syl2an 494 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
153, 6, 14syl2an 494 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
16 cdaval 8952 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
1716ancoms 469 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
18 uncom 3741 . . . 4 ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))
1917, 18syl6eq 2671 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
2015, 19breqtrrd 4651 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
214enref 7948 . . . 4 ∅ ≈ ∅
2221a1i 11 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23 cdafn 8951 . . . . 5 +𝑐 Fn (V × V)
24 fndm 5958 . . . . 5 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2523, 24ax-mp 5 . . . 4 dom +𝑐 = (V × V)
2625ndmov 6783 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27 ancom 466 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
2825ndmov 6783 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
2927, 28sylnbi 320 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
3022, 26, 293brtr4d 4655 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
3120, 30pm2.61i 176 1 (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cun 3558  cin 3559  c0 3897  {csn 4155   class class class wbr 4623   × cxp 5082  dom cdm 5084  Oncon0 5692   Fn wfn 5852  (class class class)co 6615  1𝑜c1o 7513  cen 7912   +𝑐 ccda 8949
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-1o 7520  df-er 7702  df-en 7916  df-cda 8950
This theorem is referenced by:  cdadom2  8969  cdalepw  8978  infcda  8990  alephadd  9359  gchdomtri  9411  pwxpndom  9448  gchpwdom  9452  gchhar  9461
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