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Theorem pwxpndom2 10609
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 10608 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
2 reldom 8895 . . . . . . 7 Rel ≼
32brrelex2i 5693 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 8423 . . . . . . . 8 1o = {∅}
54oveq2i 7372 . . . . . . 7 (𝐴m 1o) = (𝐴m {∅})
6 id 22 . . . . . . . 8 (𝐴 ∈ V → 𝐴 ∈ V)
7 0ex 5268 . . . . . . . . 9 ∅ ∈ V
87a1i 11 . . . . . . . 8 (𝐴 ∈ V → ∅ ∈ V)
96, 8mapsnend 8986 . . . . . . 7 (𝐴 ∈ V → (𝐴m {∅}) ≈ 𝐴)
105, 9eqbrtrid 5144 . . . . . 6 (𝐴 ∈ V → (𝐴m 1o) ≈ 𝐴)
11 ensym 8949 . . . . . 6 ((𝐴m 1o) ≈ 𝐴𝐴 ≈ (𝐴m 1o))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴m 1o))
13 map2xp 9097 . . . . . 6 (𝐴 ∈ V → (𝐴m 2o) ≈ (𝐴 × 𝐴))
14 ensym 8949 . . . . . 6 ((𝐴m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴m 2o))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴m 2o))
16 elmapi 8793 . . . . . . . . . . 11 (𝑥 ∈ (𝐴m 1o) → 𝑥:1o𝐴)
1716fdmd 6683 . . . . . . . . . 10 (𝑥 ∈ (𝐴m 1o) → dom 𝑥 = 1o)
1817adantr 482 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 1o)
19 1oex 8426 . . . . . . . . . . . . 13 1o ∈ V
2019sucid 6403 . . . . . . . . . . . 12 1o ∈ suc 1o
21 df-2o 8417 . . . . . . . . . . . 12 2o = suc 1o
2220, 21eleqtrri 2833 . . . . . . . . . . 11 1o ∈ 2o
23 1on 8428 . . . . . . . . . . . 12 1o ∈ On
2423onirri 6434 . . . . . . . . . . 11 ¬ 1o ∈ 1o
25 nelneq2 2859 . . . . . . . . . . 11 ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
2622, 24, 25mp2an 691 . . . . . . . . . 10 ¬ 2o = 1o
27 elmapi 8793 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴m 2o) → 𝑥:2o𝐴)
2827fdmd 6683 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m 2o) → dom 𝑥 = 2o)
2928adantl 483 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 2o)
3029eqeq1d 2735 . . . . . . . . . 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 3930 . . . . . . . 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 4368 . . . . 5 (ω ≼ 𝐴 → ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅)
37 djuenun 10114 . . . . 5 ((𝐴 ≈ (𝐴m 1o) ∧ (𝐴 × 𝐴) ≈ (𝐴m 2o) ∧ ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅) → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
3812, 15, 36, 37syl3anc 1372 . . . 4 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
39 omex 9587 . . . . . 6 ω ∈ V
40 ovex 7394 . . . . . 6 (𝐴m 𝑛) ∈ V
4139, 40iunex 7905 . . . . 5 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
42 1onn 8590 . . . . . . 7 1o ∈ ω
43 oveq2 7369 . . . . . . . 8 (𝑛 = 1o → (𝐴m 𝑛) = (𝐴m 1o))
4443ssiun2s 5012 . . . . . . 7 (1o ∈ ω → (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4542, 44ax-mp 5 . . . . . 6 (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
46 2onn 8592 . . . . . . 7 2o ∈ ω
47 oveq2 7369 . . . . . . . 8 (𝑛 = 2o → (𝐴m 𝑛) = (𝐴m 2o))
4847ssiun2s 5012 . . . . . . 7 (2o ∈ ω → (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4946, 48ax-mp 5 . . . . . 6 (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
5045, 49unssi 4149 . . . . 5 ((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
51 ssdomg 8946 . . . . 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 8958 . . . 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 587 . . 3 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
55 domtr 8953 . . . 4 ((𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴)) ∧ (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
5655expcom 415 . . 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 397   = wceq 1542  wcel 2107  Vcvv 3447  cun 3912  cin 3913  wss 3914  c0 4286  𝒫 cpw 4564  {csn 4590   ciun 4958   class class class wbr 5109   × cxp 5635  dom cdm 5637  suc csuc 6323  (class class class)co 7361  ωcom 7806  1oc1o 8409  2oc2o 8410  m cmap 8771  cen 8886  cdom 8887  cdju 9842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seqom 8398  df-1o 8416  df-2o 8417  df-oadd 8420  df-omul 8421  df-oexp 8422  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-oi 9454  df-har 9501  df-cnf 9606  df-dju 9845  df-card 9883
This theorem is referenced by:  pwxpndom  10610  pwdjundom  10611
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