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

Theorem pwxpndom2 10703
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 10702 . 2 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
2 reldom 8990 . . . . . . 7 Rel ≼
32brrelex2i 5746 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
4 df1o2 8512 . . . . . . . 8 1o = {∅}
54oveq2i 7442 . . . . . . 7 (𝐴m 1o) = (𝐴m {∅})
6 id 22 . . . . . . . 8 (𝐴 ∈ V → 𝐴 ∈ V)
7 0ex 5313 . . . . . . . . 9 ∅ ∈ V
87a1i 11 . . . . . . . 8 (𝐴 ∈ V → ∅ ∈ V)
96, 8mapsnend 9075 . . . . . . 7 (𝐴 ∈ V → (𝐴m {∅}) ≈ 𝐴)
105, 9eqbrtrid 5183 . . . . . 6 (𝐴 ∈ V → (𝐴m 1o) ≈ 𝐴)
11 ensym 9042 . . . . . 6 ((𝐴m 1o) ≈ 𝐴𝐴 ≈ (𝐴m 1o))
123, 10, 113syl 18 . . . . 5 (ω ≼ 𝐴𝐴 ≈ (𝐴m 1o))
13 map2xp 9186 . . . . . 6 (𝐴 ∈ V → (𝐴m 2o) ≈ (𝐴 × 𝐴))
14 ensym 9042 . . . . . 6 ((𝐴m 2o) ≈ (𝐴 × 𝐴) → (𝐴 × 𝐴) ≈ (𝐴m 2o))
153, 13, 143syl 18 . . . . 5 (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ (𝐴m 2o))
16 elmapi 8888 . . . . . . . . . . 11 (𝑥 ∈ (𝐴m 1o) → 𝑥:1o𝐴)
1716fdmd 6747 . . . . . . . . . 10 (𝑥 ∈ (𝐴m 1o) → dom 𝑥 = 1o)
1817adantr 480 . . . . . . . . 9 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 1o)
19 1oex 8515 . . . . . . . . . . . . 13 1o ∈ V
2019sucid 6468 . . . . . . . . . . . 12 1o ∈ suc 1o
21 df-2o 8506 . . . . . . . . . . . 12 2o = suc 1o
2220, 21eleqtrri 2838 . . . . . . . . . . 11 1o ∈ 2o
23 1on 8517 . . . . . . . . . . . 12 1o ∈ On
2423onirri 6499 . . . . . . . . . . 11 ¬ 1o ∈ 1o
25 nelneq2 2864 . . . . . . . . . . 11 ((1o ∈ 2o ∧ ¬ 1o ∈ 1o) → ¬ 2o = 1o)
2622, 24, 25mp2an 692 . . . . . . . . . 10 ¬ 2o = 1o
27 elmapi 8888 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴m 2o) → 𝑥:2o𝐴)
2827fdmd 6747 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m 2o) → dom 𝑥 = 2o)
2928adantl 481 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴m 1o) ∧ 𝑥 ∈ (𝐴m 2o)) → dom 𝑥 = 2o)
3029eqeq1d 2737 . . . . . . . . . 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 3979 . . . . . . . 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 4413 . . . . 5 (ω ≼ 𝐴 → ((𝐴m 1o) ∩ (𝐴m 2o)) = ∅)
37 djuenun 10209 . . . . 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 9681 . . . . . 6 ω ∈ V
40 ovex 7464 . . . . . 6 (𝐴m 𝑛) ∈ V
4139, 40iunex 7992 . . . . 5 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
42 1onn 8677 . . . . . . 7 1o ∈ ω
43 oveq2 7439 . . . . . . . 8 (𝑛 = 1o → (𝐴m 𝑛) = (𝐴m 1o))
4443ssiun2s 5053 . . . . . . 7 (1o ∈ ω → (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4542, 44ax-mp 5 . . . . . 6 (𝐴m 1o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
46 2onn 8679 . . . . . . 7 2o ∈ ω
47 oveq2 7439 . . . . . . . 8 (𝑛 = 2o → (𝐴m 𝑛) = (𝐴m 2o))
4847ssiun2s 5053 . . . . . . 7 (2o ∈ ω → (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛))
4946, 48ax-mp 5 . . . . . 6 (𝐴m 2o) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
5045, 49unssi 4201 . . . . 5 ((𝐴m 1o) ∪ (𝐴m 2o)) ⊆ 𝑛 ∈ ω (𝐴m 𝑛)
51 ssdomg 9039 . . . . 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 9051 . . . 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 9046 . . . 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 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   ciun 4996   class class class wbr 5148   × cxp 5687  dom cdm 5689  suc csuc 6388  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499  m cmap 8865  cen 8981  cdom 8982  cdju 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-har 9595  df-cnf 9700  df-dju 9939  df-card 9977
This theorem is referenced by:  pwxpndom  10704  pwdjundom  10705
  Copyright terms: Public domain W3C validator