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Theorem alephnbtwn2 9487
Description: No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephnbtwn2 ¬ ((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴))

Proof of Theorem alephnbtwn2
StepHypRef Expression
1 cardidm 9376 . . 3 (card‘(card‘𝐵)) = (card‘𝐵)
2 alephnbtwn 9486 . . 3 ((card‘(card‘𝐵)) = (card‘𝐵) → ¬ ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
31, 2ax-mp 5 . 2 ¬ ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
4 alephon 9484 . . . . . . . 8 (ℵ‘suc 𝐴) ∈ On
5 sdomdom 8524 . . . . . . . 8 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6 ondomen 9452 . . . . . . . 8 (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
74, 5, 6sylancr 590 . . . . . . 7 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8 cardid2 9370 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
97, 8syl 17 . . . . . 6 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
109ensymd 8547 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11 sdomentr 8639 . . . . 5 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
1210, 11sylan2 595 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13 alephon 9484 . . . . . 6 (ℵ‘𝐴) ∈ On
14 cardon 9361 . . . . . . 7 (card‘𝐵) ∈ On
15 onenon 9366 . . . . . . 7 ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
1614, 15ax-mp 5 . . . . . 6 (card‘𝐵) ∈ dom card
17 cardsdomel 9391 . . . . . 6 (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
1813, 16, 17mp2an 691 . . . . 5 ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
191eleq2i 2905 . . . . 5 ((ℵ‘𝐴) ∈ (card‘(card‘𝐵)) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
2018, 19bitri 278 . . . 4 ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
2112, 20sylib 221 . . 3 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ∈ (card‘𝐵))
22 ensdomtr 8641 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
239, 22mpancom 687 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
2423adantl 485 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
25 onenon 9366 . . . . . . 7 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
264, 25ax-mp 5 . . . . . 6 (ℵ‘suc 𝐴) ∈ dom card
27 cardsdomel 9391 . . . . . 6 (((card‘𝐵) ∈ On ∧ (ℵ‘suc 𝐴) ∈ dom card) → ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴))))
2814, 26, 27mp2an 691 . . . . 5 ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29 alephcard 9485 . . . . . 6 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
3029eleq2i 2905 . . . . 5 ((card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3128, 30bitri 278 . . . 4 ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3224, 31sylib 221 . . 3 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3321, 32jca 515 . 2 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
343, 33mto 200 1 ¬ ((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wcel 2114   class class class wbr 5042  dom cdm 5532  Oncon0 6169  suc csuc 6171  cfv 6334  cen 8493  cdom 8494  csdm 8495  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-card 9356  df-aleph 9357
This theorem is referenced by:  alephsucdom  9494  alephsucpw2  9526  alephgch  10085  winalim2  10107  aleph1re  15589
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