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Theorem cdacomen 9195
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 7736 . . . . 5 1𝑜 ∈ On
2 xpsneng 8210 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
31, 2mpan2 709 . . . 4 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
4 0ex 4942 . . . . 5 ∅ ∈ V
5 xpsneng 8210 . . . . 5 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
64, 5mpan2 709 . . . 4 (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7 ensym 8170 . . . . 5 ((𝐴 × {1𝑜}) ≈ 𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 ensym 8170 . . . . 5 ((𝐵 × {∅}) ≈ 𝐵𝐵 ≈ (𝐵 × {∅}))
9 incom 3948 . . . . . . 7 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜}))
10 xp01disj 7745 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅
119, 10eqtri 2782 . . . . . 6 ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅
12 cdaenun 9188 . . . . . 6 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
1311, 12mp3an3 1562 . . . . 5 ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
147, 8, 13syl2an 495 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
153, 6, 14syl2an 495 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
16 cdaval 9184 . . . . 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 3900 . . . 4 ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))
1917, 18syl6eq 2810 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
2015, 19breqtrrd 4832 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
214enref 8154 . . . 4 ∅ ≈ ∅
2221a1i 11 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23 cdafn 9183 . . . . 5 +𝑐 Fn (V × V)
24 fndm 6151 . . . . 5 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2523, 24ax-mp 5 . . . 4 dom +𝑐 = (V × V)
2625ndmov 6983 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27 ancom 465 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
2825ndmov 6983 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
2927, 28sylnbi 319 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
3022, 26, 293brtr4d 4836 . 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 383   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  cin 3714  c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  dom cdm 5266  Oncon0 5884   Fn wfn 6044  (class class class)co 6813  1𝑜c1o 7722  cen 8118   +𝑐 ccda 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-1o 7729  df-er 7911  df-en 8122  df-cda 9182
This theorem is referenced by:  cdadom2  9201  cdalepw  9210  infcda  9222  alephadd  9591  gchdomtri  9643  pwxpndom  9680  gchpwdom  9684  gchhar  9693
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