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

Theorem cdacomen 8859
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 7427 . . . . 5 1𝑜 ∈ On
2 xpsneng 7903 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
31, 2mpan2 702 . . . 4 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
4 0ex 4709 . . . . 5 ∅ ∈ V
5 xpsneng 7903 . . . . 5 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
64, 5mpan2 702 . . . 4 (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7 ensym 7864 . . . . 5 ((𝐴 × {1𝑜}) ≈ 𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 ensym 7864 . . . . 5 ((𝐵 × {∅}) ≈ 𝐵𝐵 ≈ (𝐵 × {∅}))
9 incom 3762 . . . . . . 7 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜}))
10 xp01disj 7436 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅
119, 10eqtri 2627 . . . . . 6 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅
12 cdaenun 8852 . . . . . 6 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
1311, 12mp3an3 1404 . . . . 5 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
147, 8, 13syl2an 492 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
153, 6, 14syl2an 492 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
16 cdaval 8848 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
1716ancoms 467 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
18 uncom 3714 . . . 4 ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))
1917, 18syl6eq 2655 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
2015, 19breqtrrd 4601 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
214enref 7847 . . . 4 ∅ ≈ ∅
2221a1i 11 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23 cdafn 8847 . . . . 5 +𝑐 Fn (V × V)
24 fndm 5886 . . . . 5 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2523, 24ax-mp 5 . . . 4 dom +𝑐 = (V × V)
2625ndmov 6689 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27 ancom 464 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
2825ndmov 6689 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
2927, 28sylnbi 318 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
3022, 26, 293brtr4d 4605 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
3120, 30pm2.61i 174 1 (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cun 3533  cin 3534  c0 3869  {csn 4120   class class class wbr 4573   × cxp 5022  dom cdm 5024  Oncon0 5622   Fn wfn 5781  (class class class)co 6523  1𝑜c1o 7413  cen 7811   +𝑐 ccda 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-1o 7420  df-er 7602  df-en 7815  df-cda 8846
This theorem is referenced by:  cdadom2  8865  cdalepw  8874  infcda  8886  alephadd  9251  gchdomtri  9303  pwxpndom  9340  gchpwdom  9344  gchhar  9353
  Copyright terms: Public domain W3C validator