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Theorem pwxpndom2 10076
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 10075 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
2 reldom 8498 . . . . . . 7 Rel ≼
32brrelex2i 5573 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 8099 . . . . . . . 8 1o = {∅}
54oveq2i 7146 . . . . . . 7 (𝐴m 1o) = (𝐴m {∅})
6 id 22 . . . . . . . 8 (𝐴 ∈ V → 𝐴 ∈ V)
7 0ex 5175 . . . . . . . . 9 ∅ ∈ V
87a1i 11 . . . . . . . 8 (𝐴 ∈ V → ∅ ∈ V)
96, 8mapsnend 8571 . . . . . . 7 (𝐴 ∈ V → (𝐴m {∅}) ≈ 𝐴)
105, 9eqbrtrid 5065 . . . . . 6 (𝐴 ∈ V → (𝐴m 1o) ≈ 𝐴)
11 ensym 8541 . . . . . 6 ((𝐴m 1o) ≈ 𝐴𝐴 ≈ (𝐴m 1o))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴m 1o))
13 map2xp 8671 . . . . . 6 (𝐴 ∈ V → (𝐴m 2o) ≈ (𝐴 × 𝐴))
14 ensym 8541 . . . . . 6 ((𝐴m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴m 2o))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴m 2o))
16 elmapi 8411 . . . . . . . . . . 11 (𝑥 ∈ (𝐴m 1o) → 𝑥:1o𝐴)
1716fdmd 6497 . . . . . . . . . 10 (𝑥 ∈ (𝐴m 1o) → dom 𝑥 = 1o)
1817adantr 484 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 1o)
19 1oex 8093 . . . . . . . . . . . . 13 1o ∈ V
2019sucid 6238 . . . . . . . . . . . 12 1o ∈ suc 1o
21 df-2o 8086 . . . . . . . . . . . 12 2o = suc 1o
2220, 21eleqtrri 2889 . . . . . . . . . . 11 1o ∈ 2o
23 1on 8092 . . . . . . . . . . . 12 1o ∈ On
2423onirri 6265 . . . . . . . . . . 11 ¬ 1o ∈ 1o
25 nelneq2 2915 . . . . . . . . . . 11 ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
2622, 24, 25mp2an 691 . . . . . . . . . 10 ¬ 2o = 1o
27 elmapi 8411 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴m 2o) → 𝑥:2o𝐴)
2827fdmd 6497 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m 2o) → dom 𝑥 = 2o)
2928adantl 485 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 2o)
3029eqeq1d 2800 . . . . . . . . . 10 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → (dom 𝑥 = 1o ↔ 2o = 1o))
3126, 30mtbiri 330 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → ¬ dom 𝑥 = 1o)
3218, 31pm2.65i 197 . . . . . . . 8 ¬ (𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o))
33 elin 3897 . . . . . . . 8 (𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o)) ↔ (𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)))
3432, 33mtbir 326 . . . . . . 7 ¬ 𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o))
3534a1i 11 . . . . . 6 (ω ≼ 𝐴 → ¬ 𝑥 ∈ ((𝐴m 1o) ∩ (𝐴m 2o)))
3635eq0rdv 4312 . . . . 5 (ω ≼ 𝐴 → ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅)
37 djuenun 9581 . . . . 5 ((𝐴 ≈ (𝐴m 1o) ∧ (𝐴 × 𝐴) ≈ (𝐴m 2o) ∧ ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅) → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
3812, 15, 36, 37syl3anc 1368 . . . 4 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
39 omex 9090 . . . . . 6 ω ∈ V
40 ovex 7168 . . . . . 6 (𝐴m 𝑛) ∈ V
4139, 40iunex 7651 . . . . 5 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
42 1onn 8248 . . . . . . 7 1o ∈ ω
43 oveq2 7143 . . . . . . . 8 (𝑛 = 1o → (𝐴m 𝑛) = (𝐴m 1o))
4443ssiun2s 4935 . . . . . . 7 (1o ∈ ω → (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4542, 44ax-mp 5 . . . . . 6 (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
46 2onn 8249 . . . . . . 7 2o ∈ ω
47 oveq2 7143 . . . . . . . 8 (𝑛 = 2o → (𝐴m 𝑛) = (𝐴m 2o))
4847ssiun2s 4935 . . . . . . 7 (2o ∈ ω → (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4946, 48ax-mp 5 . . . . . 6 (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
5045, 49unssi 4112 . . . . 5 ((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
51 ssdomg 8538 . . . . 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 8550 . . . 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 589 . . 3 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
55 domtr 8545 . . . 4 ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
5655expcom 417 . . 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 201 1 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  cun 3879  cin 3880  wss 3881  c0 4243  𝒫 cpw 4497  {csn 4525   ciun 4881   class class class wbr 5030   × cxp 5517  dom cdm 5519  suc csuc 6161  (class class class)co 7135  ωcom 7560  1oc1o 8078  2oc2o 8079  m cmap 8389  cen 8489  cdom 8490  cdju 9311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-har 9005  df-cnf 9109  df-dju 9314  df-card 9352
This theorem is referenced by:  pwxpndom  10077  pwdjundom  10078
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