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Theorem pwxpndom2 9689
Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 18-Jul-2022.)
Assertion
Ref Expression
pwxpndom2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))

Proof of Theorem pwxpndom2
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwfseq 9688 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
2 reldom 8115 . . . . . . 7 Rel ≼
32brrelex2i 5299 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 7726 . . . . . . . 8 1𝑜 = {∅}
54oveq2i 6804 . . . . . . 7 (𝐴𝑚 1𝑜) = (𝐴𝑚 {∅})
6 id 22 . . . . . . . 8 (𝐴 ∈ V → 𝐴 ∈ V)
7 0ex 4924 . . . . . . . . 9 ∅ ∈ V
87a1i 11 . . . . . . . 8 (𝐴 ∈ V → ∅ ∈ V)
96, 8mapsnend 8188 . . . . . . 7 (𝐴 ∈ V → (𝐴𝑚 {∅}) ≈ 𝐴)
105, 9syl5eqbr 4821 . . . . . 6 (𝐴 ∈ V → (𝐴𝑚 1𝑜) ≈ 𝐴)
11 ensym 8158 . . . . . 6 ((𝐴𝑚 1𝑜) ≈ 𝐴𝐴 ≈ (𝐴𝑚 1𝑜))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴𝑚 1𝑜))
13 map2xp 8286 . . . . . 6 (𝐴 ∈ V → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
14 ensym 8158 . . . . . 6 ((𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴𝑚 2𝑜))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴𝑚 2𝑜))
16 elmapi 8031 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝑚 1𝑜) → 𝑥:1𝑜𝐴)
17 fdm 6191 . . . . . . . . . . 11 (𝑥:1𝑜𝐴 → dom 𝑥 = 1𝑜)
1816, 17syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝑚 1𝑜) → dom 𝑥 = 1𝑜)
1918adantr 466 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜)) → dom 𝑥 = 1𝑜)
20 1oex 7721 . . . . . . . . . . . . 13 1𝑜 ∈ V
2120sucid 5947 . . . . . . . . . . . 12 1𝑜 ∈ suc 1𝑜
22 df-2o 7714 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
2321, 22eleqtrri 2849 . . . . . . . . . . 11 1𝑜 ∈ 2𝑜
24 1on 7720 . . . . . . . . . . . 12 1𝑜 ∈ On
2524onirri 5977 . . . . . . . . . . 11 ¬ 1𝑜 ∈ 1𝑜
26 nelneq2 2875 . . . . . . . . . . 11 ((1𝑜 ∈ 2𝑜 ∧ ¬ 1𝑜 ∈ 1𝑜) → ¬ 2𝑜 = 1𝑜)
2723, 25, 26mp2an 664 . . . . . . . . . 10 ¬ 2𝑜 = 1𝑜
28 elmapi 8031 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴𝑚 2𝑜) → 𝑥:2𝑜𝐴)
29 fdm 6191 . . . . . . . . . . . . 13 (𝑥:2𝑜𝐴 → dom 𝑥 = 2𝑜)
3028, 29syl 17 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴𝑚 2𝑜) → dom 𝑥 = 2𝑜)
3130adantl 467 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜)) → dom 𝑥 = 2𝑜)
3231eqeq1d 2773 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜)) → (dom 𝑥 = 1𝑜 ↔ 2𝑜 = 1𝑜))
3327, 32mtbiri 316 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜)) → ¬ dom 𝑥 = 1𝑜)
3419, 33pm2.65i 185 . . . . . . . 8 ¬ (𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜))
35 elin 3947 . . . . . . . 8 (𝑥 ∈ ((𝐴𝑚 1𝑜) ∩ (𝐴𝑚 2𝑜)) ↔ (𝑥 ∈ (𝐴𝑚 1𝑜) ∧ 𝑥 ∈ (𝐴𝑚 2𝑜)))
3634, 35mtbir 312 . . . . . . 7 ¬ 𝑥 ∈ ((𝐴𝑚 1𝑜) ∩ (𝐴𝑚 2𝑜))
3736a1i 11 . . . . . 6 (ω ≼ 𝐴 → ¬ 𝑥 ∈ ((𝐴𝑚 1𝑜) ∩ (𝐴𝑚 2𝑜)))
3837eq0rdv 4123 . . . . 5 (ω ≼ 𝐴 → ((𝐴𝑚 1𝑜) ∩ (𝐴𝑚 2𝑜)) = ∅)
39 cdaenun 9198 . . . . 5 ((𝐴 ≈ (𝐴𝑚 1𝑜) ∧ (𝐴 × 𝐴) ≈ (𝐴𝑚 2𝑜) ∧ ((𝐴𝑚 1𝑜) ∩ (𝐴𝑚 2𝑜)) = ∅) → (𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)))
4012, 15, 38, 39syl3anc 1476 . . . 4 (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)))
41 omex 8704 . . . . . 6 ω ∈ V
42 ovex 6823 . . . . . 6 (𝐴𝑚 𝑛) ∈ V
4341, 42iunex 7294 . . . . 5 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
44 1onn 7873 . . . . . . 7 1𝑜 ∈ ω
45 oveq2 6801 . . . . . . . 8 (𝑛 = 1𝑜 → (𝐴𝑚 𝑛) = (𝐴𝑚 1𝑜))
4645ssiun2s 4698 . . . . . . 7 (1𝑜 ∈ ω → (𝐴𝑚 1𝑜) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛))
4744, 46ax-mp 5 . . . . . 6 (𝐴𝑚 1𝑜) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛)
48 2onn 7874 . . . . . . 7 2𝑜 ∈ ω
49 oveq2 6801 . . . . . . . 8 (𝑛 = 2𝑜 → (𝐴𝑚 𝑛) = (𝐴𝑚 2𝑜))
5049ssiun2s 4698 . . . . . . 7 (2𝑜 ∈ ω → (𝐴𝑚 2𝑜) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛))
5148, 50ax-mp 5 . . . . . 6 (𝐴𝑚 2𝑜) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛)
5247, 51unssi 3939 . . . . 5 ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛)
53 ssdomg 8155 . . . . 5 ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V → (((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ⊆ 𝑛 ∈ ω (𝐴𝑚 𝑛) → ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)))
5443, 52, 53mp2 9 . . . 4 ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)
55 endomtr 8167 . . . 4 (((𝐴 +𝑐 (𝐴 × 𝐴)) ≈ ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ∧ ((𝐴𝑚 1𝑜) ∪ (𝐴𝑚 2𝑜)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)) → (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
5640, 54, 55sylancl 566 . . 3 (ω ≼ 𝐴 → (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
57 domtr 8162 . . . 4 ((𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) ∧ (𝐴 +𝑐 (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
5857expcom 398 . . 3 ((𝐴 +𝑐 (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛) → (𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
5956, 58syl 17 . 2 (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
601, 59mtod 189 1 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   ciun 4654   class class class wbr 4786   × cxp 5247  dom cdm 5249  suc csuc 5868  wf 6027  (class class class)co 6793  ωcom 7212  1𝑜c1o 7706  2𝑜c2o 7707  𝑚 cmap 8009  cen 8106  cdom 8107   +𝑐 ccda 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-har 8619  df-cnf 8723  df-card 8965  df-cda 9192
This theorem is referenced by:  pwxpndom  9690  pwcdandom  9691
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