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Theorem alephnbtwn 10109
Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephnbtwn ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))

Proof of Theorem alephnbtwn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 alephon 10107 . . . . . . . 8 (ℵ‘𝐴) ∈ On
2 id 22 . . . . . . . . . 10 ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3 cardon 9982 . . . . . . . . . 10 (card‘𝐵) ∈ On
42, 3eqeltrrdi 2848 . . . . . . . . 9 ((card‘𝐵) = 𝐵𝐵 ∈ On)
5 onenon 9987 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((card‘𝐵) = 𝐵𝐵 ∈ dom card)
7 cardsdomel 10012 . . . . . . . 8 (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
81, 6, 7sylancr 587 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9 eleq2 2828 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
108, 9bitrd 279 . . . . . 6 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
1110adantl 481 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12 alephsuc 10106 . . . . . . . . . . 11 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13 onenon 9987 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14 harval2 10035 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ dom card → (har‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
151, 13, 14mp2b 10 . . . . . . . . . . 11 (har‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}
1612, 15eqtrdi 2791 . . . . . . . . . 10 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
1716eleq2d 2825 . . . . . . . . 9 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
1817biimpd 229 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19 breq2 5152 . . . . . . . . 9 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
2019onnminsb 7819 . . . . . . . 8 (𝐵 ∈ On → (𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
2118, 20sylan9 507 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
2221con2d 134 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
234, 22sylan2 593 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
2411, 23sylbird 260 . . . 4 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25 imnan 399 . . . 4 (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2624, 25sylib 218 . . 3 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2726ex 412 . 2 (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
28 n0i 4346 . . . . . . 7 (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29 alephfnon 10103 . . . . . . . . . 10 ℵ Fn On
3029fndmi 6673 . . . . . . . . 9 dom ℵ = On
3130eleq2i 2831 . . . . . . . 8 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
32 ndmfv 6942 . . . . . . . 8 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
3331, 32sylnbir 331 . . . . . . 7 (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
3428, 33nsyl2 141 . . . . . 6 (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
35 onsucb 7837 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3634, 35sylibr 234 . . . . 5 (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
3736adantl 481 . . . 4 (((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
3837con3i 154 . . 3 𝐴 ∈ On → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
3938a1d 25 . 2 𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
4027, 39pm2.61i 182 1 ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  c0 4339   cint 4951   class class class wbr 5148  dom cdm 5689  Oncon0 6386  suc csuc 6388  cfv 6563  csdm 8983  harchar 9594  cardccrd 9973  cale 9974
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-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-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978
This theorem is referenced by:  alephnbtwn2  10110
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