MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwxpndom2 Structured version   Visualization version   GIF version

Theorem pwxpndom2 10705
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 10704 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
2 reldom 8991 . . . . . . 7 Rel ≼
32brrelex2i 5742 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 8513 . . . . . . . 8 1o = {∅}
54oveq2i 7442 . . . . . . 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 9076 . . . . . . 7 (𝐴 ∈ V → (𝐴m {∅}) ≈ 𝐴)
105, 9eqbrtrid 5178 . . . . . 6 (𝐴 ∈ V → (𝐴m 1o) ≈ 𝐴)
11 ensym 9043 . . . . . 6 ((𝐴m 1o) ≈ 𝐴𝐴 ≈ (𝐴m 1o))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴m 1o))
13 map2xp 9187 . . . . . 6 (𝐴 ∈ V → (𝐴m 2o) ≈ (𝐴 × 𝐴))
14 ensym 9043 . . . . . 6 ((𝐴m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴m 2o))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴m 2o))
16 elmapi 8889 . . . . . . . . . . 11 (𝑥 ∈ (𝐴m 1o) → 𝑥:1o𝐴)
1716fdmd 6746 . . . . . . . . . 10 (𝑥 ∈ (𝐴m 1o) → dom 𝑥 = 1o)
1817adantr 480 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 1o)
19 1oex 8516 . . . . . . . . . . . . 13 1o ∈ V
2019sucid 6466 . . . . . . . . . . . 12 1o ∈ suc 1o
21 df-2o 8507 . . . . . . . . . . . 12 2o = suc 1o
2220, 21eleqtrri 2840 . . . . . . . . . . 11 1o ∈ 2o
23 1on 8518 . . . . . . . . . . . 12 1o ∈ On
2423onirri 6497 . . . . . . . . . . 11 ¬ 1o ∈ 1o
25 nelneq2 2866 . . . . . . . . . . 11 ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
2622, 24, 25mp2an 692 . . . . . . . . . 10 ¬ 2o = 1o
27 elmapi 8889 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴m 2o) → 𝑥:2o𝐴)
2827fdmd 6746 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m 2o) → dom 𝑥 = 2o)
2928adantl 481 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 2o)
3029eqeq1d 2739 . . . . . . . . . 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 194 . . . . . . . 8 ¬ (𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o))
33 elin 3967 . . . . . . . 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 4407 . . . . 5 (ω ≼ 𝐴 → ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅)
37 djuenun 10211 . . . . 5 ((𝐴 ≈ (𝐴m 1o) ∧ (𝐴 × 𝐴) ≈ (𝐴m 2o) ∧ ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅) → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
3812, 15, 36, 37syl3anc 1373 . . . 4 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≈ ((𝐴m 1o) ∪ (𝐴m 2o)))
39 omex 9683 . . . . . 6 ω ∈ V
40 ovex 7464 . . . . . 6 (𝐴m 𝑛) ∈ V
4139, 40iunex 7993 . . . . 5 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
42 1onn 8678 . . . . . . 7 1o ∈ ω
43 oveq2 7439 . . . . . . . 8 (𝑛 = 1o → (𝐴m 𝑛) = (𝐴m 1o))
4443ssiun2s 5048 . . . . . . 7 (1o ∈ ω → (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4542, 44ax-mp 5 . . . . . 6 (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
46 2onn 8680 . . . . . . 7 2o ∈ ω
47 oveq2 7439 . . . . . . . 8 (𝑛 = 2o → (𝐴m 𝑛) = (𝐴m 2o))
4847ssiun2s 5048 . . . . . . 7 (2o ∈ ω → (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4946, 48ax-mp 5 . . . . . 6 (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
5045, 49unssi 4191 . . . . 5 ((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
51 ssdomg 9040 . . . . 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 9052 . . . 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 586 . . 3 (ω ≼ 𝐴 → (𝐴 ⊔ (𝐴 × 𝐴)) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
55 domtr 9047 . . . 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 198 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 2108  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   ciun 4991   class class class wbr 5143   × cxp 5683  dom cdm 5685  suc csuc 6386  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  m cmap 8866  cen 8982  cdom 8983  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-har 9597  df-cnf 9702  df-dju 9941  df-card 9979
This theorem is referenced by:  pwxpndom  10706  pwdjundom  10707
  Copyright terms: Public domain W3C validator