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Theorem alephnbtwn 9104
 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 9102 . . . . . . . 8 (ℵ‘𝐴) ∈ On
2 id 22 . . . . . . . . . 10 ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3 cardon 8980 . . . . . . . . . 10 (card‘𝐵) ∈ On
42, 3syl6eqelr 2848 . . . . . . . . 9 ((card‘𝐵) = 𝐵𝐵 ∈ On)
5 onenon 8985 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((card‘𝐵) = 𝐵𝐵 ∈ dom card)
7 cardsdomel 9010 . . . . . . . 8 (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
81, 6, 7sylancr 698 . . . . . . 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 268 . . . . . 6 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
1110adantl 473 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12 alephsuc 9101 . . . . . . . . . . 11 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13 onenon 8985 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14 harval2 9033 . . . . . . . . . . . 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 2810 . . . . . . . . . 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 219 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19 breq2 4808 . . . . . . . . 9 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
2019onnminsb 7170 . . . . . . . 8 (𝐵 ∈ On → (𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
2118, 20sylan9 692 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
2221con2d 129 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
234, 22sylan2 492 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
2411, 23sylbird 250 . . . 4 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25 imnan 437 . . . 4 (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2624, 25sylib 208 . . 3 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2726ex 449 . 2 (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
28 n0i 4063 . . . . . . 7 (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29 alephfnon 9098 . . . . . . . . . 10 ℵ Fn On
30 fndm 6151 . . . . . . . . . 10 (ℵ Fn On → dom ℵ = On)
3129, 30ax-mp 5 . . . . . . . . 9 dom ℵ = On
3231eleq2i 2831 . . . . . . . 8 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
33 ndmfv 6380 . . . . . . . 8 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
3432, 33sylnbir 320 . . . . . . 7 (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
3528, 34nsyl2 142 . . . . . 6 (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
36 sucelon 7183 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3735, 36sylibr 224 . . . . 5 (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
3837adantl 473 . . . 4 (((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
3938con3i 150 . . 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 176 1 ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054  ∅c0 4058  ∩ cint 4627   class class class wbr 4804  dom cdm 5266  Oncon0 5884  suc csuc 5886   Fn wfn 6044  ‘cfv 6049   ≺ csdm 8122  harchar 8628  cardccrd 8971  ℵcale 8972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-oi 8582  df-har 8630  df-card 8975  df-aleph 8976 This theorem is referenced by:  alephnbtwn2  9105
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