Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdacomen Structured version   Visualization version   GIF version

Theorem cdacomen 9318
 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 7833 . . . . 5 1o ∈ On
2 xpsneng 8314 . . . . 5 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
31, 2mpan2 682 . . . 4 (𝐴 ∈ V → (𝐴 × {1o}) ≈ 𝐴)
4 0ex 5014 . . . . 5 ∅ ∈ V
5 xpsneng 8314 . . . . 5 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
64, 5mpan2 682 . . . 4 (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7 ensym 8271 . . . . 5 ((𝐴 × {1o}) ≈ 𝐴𝐴 ≈ (𝐴 × {1o}))
8 ensym 8271 . . . . 5 ((𝐵 × {∅}) ≈ 𝐵𝐵 ≈ (𝐵 × {∅}))
9 incom 4032 . . . . . . 7 ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1o}))
10 xp01disj 7843 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐴 × {1o})) = ∅
119, 10eqtri 2849 . . . . . 6 ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ∅
12 cdaenun 9311 . . . . . 6 ((𝐴 ≈ (𝐴 × {1o}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
1311, 12mp3an3 1578 . . . . 5 ((𝐴 ≈ (𝐴 × {1o}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
147, 8, 13syl2an 589 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
153, 6, 14syl2an 589 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
16 cdaval 9307 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1o})))
1716ancoms 452 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1o})))
18 uncom 3984 . . . 4 ((𝐵 × {∅}) ∪ (𝐴 × {1o})) = ((𝐴 × {1o}) ∪ (𝐵 × {∅}))
1917, 18syl6eq 2877 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
2015, 19breqtrrd 4901 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
214enref 8255 . . . 4 ∅ ≈ ∅
2221a1i 11 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23 cdafn 9306 . . . . 5 +𝑐 Fn (V × V)
24 fndm 6223 . . . . 5 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2523, 24ax-mp 5 . . . 4 dom +𝑐 = (V × V)
2625ndmov 7078 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27 ancom 454 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
2825ndmov 7078 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
2927, 28sylnbi 322 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
3022, 26, 293brtr4d 4905 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
3120, 30pm2.61i 177 1 (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∪ cun 3796   ∩ cin 3797  ∅c0 4144  {csn 4397   class class class wbr 4873   × cxp 5340  dom cdm 5342  Oncon0 5963   Fn wfn 6118  (class class class)co 6905  1oc1o 7819   ≈ cen 8219   +𝑐 ccda 9304 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-1o 7826  df-er 8009  df-en 8223  df-cda 9305 This theorem is referenced by:  cdadom2  9324  cdalepw  9333  infcda  9345  alephadd  9714  gchdomtri  9766  pwxpndom  9803  gchpwdom  9807  gchhar  9816
 Copyright terms: Public domain W3C validator