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Theorem alephnbtwn 9485
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 9483 . . . . . . . 8 (ℵ‘𝐴) ∈ On
2 id 22 . . . . . . . . . 10 ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3 cardon 9361 . . . . . . . . . 10 (card‘𝐵) ∈ On
42, 3syl6eqelr 2919 . . . . . . . . 9 ((card‘𝐵) = 𝐵𝐵 ∈ On)
5 onenon 9366 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((card‘𝐵) = 𝐵𝐵 ∈ dom card)
7 cardsdomel 9391 . . . . . . . 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 2898 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
108, 9bitrd 280 . . . . . 6 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
1110adantl 482 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12 alephsuc 9482 . . . . . . . . . . 11 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13 onenon 9366 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14 harval2 9414 . . . . . . . . . . . 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, 15syl6eq 2869 . . . . . . . . . 10 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
1716eleq2d 2895 . . . . . . . . 9 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
1817biimpd 230 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19 breq2 5061 . . . . . . . . 9 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
2019onnminsb 7508 . . . . . . . 8 (𝐵 ∈ On → (𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
2118, 20sylan9 508 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
2221con2d 136 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
234, 22sylan2 592 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
2411, 23sylbird 261 . . . 4 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25 imnan 400 . . . 4 (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2624, 25sylib 219 . . 3 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2726ex 413 . 2 (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
28 n0i 4296 . . . . . . 7 (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29 alephfnon 9479 . . . . . . . . . 10 ℵ Fn On
30 fndm 6448 . . . . . . . . . 10 (ℵ Fn On → dom ℵ = On)
3129, 30ax-mp 5 . . . . . . . . 9 dom ℵ = On
3231eleq2i 2901 . . . . . . . 8 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
33 ndmfv 6693 . . . . . . . 8 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
3432, 33sylnbir 332 . . . . . . 7 (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
3528, 34nsyl2 143 . . . . . 6 (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
36 sucelon 7521 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3735, 36sylibr 235 . . . . 5 (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
3837adantl 482 . . . 4 (((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
3938con3i 157 . . 3 𝐴 ∈ On → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
4039a1d 25 . 2 𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
4127, 40pm2.61i 183 1 ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  c0 4288   cint 4867   class class class wbr 5057  dom cdm 5548  Oncon0 6184  suc csuc 6186   Fn wfn 6343  cfv 6348  csdm 8496  harchar 9008  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357
This theorem is referenced by:  alephnbtwn2  9486
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