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Theorem alephnbtwn 9496
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 9494 . . . . . . . 8 (ℵ‘𝐴) ∈ On
2 id 22 . . . . . . . . . 10 ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3 cardon 9372 . . . . . . . . . 10 (card‘𝐵) ∈ On
42, 3eqeltrrdi 2922 . . . . . . . . 9 ((card‘𝐵) = 𝐵𝐵 ∈ On)
5 onenon 9377 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((card‘𝐵) = 𝐵𝐵 ∈ dom card)
7 cardsdomel 9402 . . . . . . . 8 (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
81, 6, 7sylancr 589 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9 eleq2 2901 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
108, 9bitrd 281 . . . . . 6 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
1110adantl 484 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12 alephsuc 9493 . . . . . . . . . . 11 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13 onenon 9377 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14 harval2 9425 . . . . . . . . . . . 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 2872 . . . . . . . . . 10 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
1716eleq2d 2898 . . . . . . . . 9 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
1817biimpd 231 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19 breq2 5069 . . . . . . . . 9 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
2019onnminsb 7518 . . . . . . . 8 (𝐵 ∈ On → (𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
2118, 20sylan9 510 . . . . . . 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 594 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
2411, 23sylbird 262 . . . 4 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25 imnan 402 . . . 4 (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2624, 25sylib 220 . . 3 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2726ex 415 . 2 (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
28 n0i 4298 . . . . . . 7 (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29 alephfnon 9490 . . . . . . . . . 10 ℵ Fn On
30 fndm 6454 . . . . . . . . . 10 (ℵ Fn On → dom ℵ = On)
3129, 30ax-mp 5 . . . . . . . . 9 dom ℵ = On
3231eleq2i 2904 . . . . . . . 8 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
33 ndmfv 6699 . . . . . . . 8 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
3432, 33sylnbir 333 . . . . . . 7 (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
3528, 34nsyl2 143 . . . . . 6 (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
36 sucelon 7531 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3735, 36sylibr 236 . . . . 5 (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
3837adantl 484 . . . 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 184 1 ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  {crab 3142  c0 4290   cint 4875   class class class wbr 5065  dom cdm 5554  Oncon0 6190  suc csuc 6192   Fn wfn 6349  cfv 6354  csdm 8507  harchar 9019  cardccrd 9363  cale 9364
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-oi 8973  df-har 9021  df-card 9367  df-aleph 9368
This theorem is referenced by:  alephnbtwn2  9497
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