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Theorem alephnbtwn 8751
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 8749 . . . . . . . 8 (ℵ‘𝐴) ∈ On
2 id 22 . . . . . . . . . 10 ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3 cardon 8627 . . . . . . . . . 10 (card‘𝐵) ∈ On
42, 3syl6eqelr 2693 . . . . . . . . 9 ((card‘𝐵) = 𝐵𝐵 ∈ On)
5 onenon 8632 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((card‘𝐵) = 𝐵𝐵 ∈ dom card)
7 cardsdomel 8657 . . . . . . . 8 (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
81, 6, 7sylancr 693 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9 eleq2 2673 . . . . . . 7 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
108, 9bitrd 266 . . . . . 6 ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
1110adantl 480 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12 alephsuc 8748 . . . . . . . . . . 11 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13 onenon 8632 . . . . . . . . . . . 12 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14 harval2 8680 . . . . . . . . . . . 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 2656 . . . . . . . . . 10 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
1716eleq2d 2669 . . . . . . . . 9 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
1817biimpd 217 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19 breq2 4578 . . . . . . . . 9 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
2019onnminsb 6870 . . . . . . . 8 (𝐵 ∈ On → (𝐵 {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
2118, 20sylan9 686 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
2221con2d 127 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
234, 22sylan2 489 . . . . 5 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
2411, 23sylbird 248 . . . 4 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25 imnan 436 . . . 4 (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2624, 25sylib 206 . . 3 ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
2726ex 448 . 2 (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴))))
28 n0i 3875 . . . . . . 7 (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29 alephfnon 8745 . . . . . . . . . 10 ℵ Fn On
30 fndm 5887 . . . . . . . . . 10 (ℵ Fn On → dom ℵ = On)
3129, 30ax-mp 5 . . . . . . . . 9 dom ℵ = On
3231eleq2i 2676 . . . . . . . 8 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
33 ndmfv 6110 . . . . . . . 8 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
3432, 33sylnbir 319 . . . . . . 7 (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
3528, 34nsyl2 140 . . . . . 6 (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
36 sucelon 6883 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3735, 36sylibr 222 . . . . 5 (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
3837adantl 480 . . . 4 (((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
3938con3i 148 . . 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 174 1 ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {crab 2896  c0 3870   cint 4401   class class class wbr 4574  dom cdm 5025  Oncon0 5623  suc csuc 5625   Fn wfn 5782  cfv 5787  csdm 7814  harchar 8318  cardccrd 8618  cale 8619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-oi 8272  df-har 8320  df-card 8622  df-aleph 8623
This theorem is referenced by:  alephnbtwn2  8752
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