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Theorem pwxpndom2 10664
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 ⊔ (𝐴 × 𝐴)))

Proof of Theorem pwxpndom2
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwfseq 10663 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
2 reldom 8949 . . . . . . 7 Rel ≼
32brrelex2i 5733 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 8477 . . . . . . . 8 1o = {∅}
54oveq2i 7423 . . . . . . 7 (𝐴m 1o) = (𝐴m {∅})
6 id 22 . . . . . . . 8 (𝐴 ∈ V → 𝐴 ∈ V)
7 0ex 5307 . . . . . . . . 9 ∅ ∈ V
87a1i 11 . . . . . . . 8 (𝐴 ∈ V → ∅ ∈ V)
96, 8mapsnend 9040 . . . . . . 7 (𝐴 ∈ V → (𝐴m {∅}) ≈ 𝐴)
105, 9eqbrtrid 5183 . . . . . 6 (𝐴 ∈ V → (𝐴m 1o) ≈ 𝐴)
11 ensym 9003 . . . . . 6 ((𝐴m 1o) ≈ 𝐴𝐴 ≈ (𝐴m 1o))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴m 1o))
13 map2xp 9151 . . . . . 6 (𝐴 ∈ V → (𝐴m 2o) ≈ (𝐴 × 𝐴))
14 ensym 9003 . . . . . 6 ((𝐴m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴m 2o))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴m 2o))
16 elmapi 8847 . . . . . . . . . . 11 (𝑥 ∈ (𝐴m 1o) → 𝑥:1o𝐴)
1716fdmd 6728 . . . . . . . . . 10 (𝑥 ∈ (𝐴m 1o) → dom 𝑥 = 1o)
1817adantr 480 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 1o)
19 1oex 8480 . . . . . . . . . . . . 13 1o ∈ V
2019sucid 6446 . . . . . . . . . . . 12 1o ∈ suc 1o
21 df-2o 8471 . . . . . . . . . . . 12 2o = suc 1o
2220, 21eleqtrri 2831 . . . . . . . . . . 11 1o ∈ 2o
23 1on 8482 . . . . . . . . . . . 12 1o ∈ On
2423onirri 6477 . . . . . . . . . . 11 ¬ 1o ∈ 1o
25 nelneq2 2857 . . . . . . . . . . 11 ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
2622, 24, 25mp2an 689 . . . . . . . . . 10 ¬ 2o = 1o
27 elmapi 8847 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴m 2o) → 𝑥:2o𝐴)
2827fdmd 6728 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m 2o) → dom 𝑥 = 2o)
2928adantl 481 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 2o)
3029eqeq1d 2733 . . . . . . . . . 10 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → (dom 𝑥 = 1o ↔ 2o = 1o))
3126, 30mtbiri 327 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → ¬ dom 𝑥 = 1o)
3218, 31pm2.65i 193 . . . . . . . 8 ¬ (𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o))
33 elin 3964 . . . . . . . 8 (𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o)) ↔ (𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)))
3432, 33mtbir 323 . . . . . . 7 ¬ 𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o))
3534a1i 11 . . . . . 6 (ω ≼ 𝐴 → ¬ 𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o)))
3635eq0rdv 4404 . . . . 5 (ω ≼ 𝐴 → ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅)
37 djuenun 10169 . . . . 5 ((𝐴 ≈ (𝐴m 1o) ∧ (𝐴 × 𝐴) ≈ (𝐴m 2o) ∧ ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅) → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
3812, 15, 36, 37syl3anc 1370 . . . 4 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
39 omex 9642 . . . . . 6 ω ∈ V
40 ovex 7445 . . . . . 6 (𝐴m 𝑛) ∈ V
4139, 40iunex 7959 . . . . 5 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
42 1onn 8643 . . . . . . 7 1o ∈ ω
43 oveq2 7420 . . . . . . . 8 (𝑛 = 1o → (𝐴m 𝑛) = (𝐴m 1o))
4443ssiun2s 5051 . . . . . . 7 (1o ∈ ω → (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4542, 44ax-mp 5 . . . . . 6 (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
46 2onn 8645 . . . . . . 7 2o ∈ ω
47 oveq2 7420 . . . . . . . 8 (𝑛 = 2o → (𝐴m 𝑛) = (𝐴m 2o))
4847ssiun2s 5051 . . . . . . 7 (2o ∈ ω → (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4946, 48ax-mp 5 . . . . . 6 (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
5045, 49unssi 4185 . . . . 5 ((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
51 ssdomg 9000 . . . . 5 ( 𝑛 ∈ ω (𝐴m 𝑛) ∈ V → (((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛) → ((𝐴m 1o) ∪ (𝐴m 2o)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)))
5241, 50, 51mp2 9 . . . 4 ((𝐴m 1o) ∪ (𝐴m 2o)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)
53 endomtr 9012 . . . 4 (((𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)) ∧ ((𝐴m 1o) ∪ (𝐴m 2o)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
5438, 52, 53sylancl 585 . . 3 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
55 domtr 9007 . . . 4 ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
5655expcom 413 . . 3 ((𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛) → (𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
5754, 56syl 17 . 2 (ω ≼ 𝐴 → (𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
581, 57mtod 197 1 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cun 3946  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   ciun 4997   class class class wbr 5148   × cxp 5674  dom cdm 5676  suc csuc 6366  (class class class)co 7412  ωcom 7859  1oc1o 8463  2oc2o 8464  m cmap 8824  cen 8940  cdom 8941  cdju 9897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-seqom 8452  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-oi 9509  df-har 9556  df-cnf 9661  df-dju 9900  df-card 9938
This theorem is referenced by:  pwxpndom  10665  pwdjundom  10666
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