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Theorem alephnbtwn2 8840
 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 8730 . . 3 (card‘(card‘𝐵)) = (card‘𝐵)
2 alephnbtwn 8839 . . 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 8837 . . . . . . . 8 (ℵ‘suc 𝐴) ∈ On
5 sdomdom 7928 . . . . . . . 8 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6 ondomen 8805 . . . . . . . 8 (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
74, 5, 6sylancr 694 . . . . . . 7 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8 cardid2 8724 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
97, 8syl 17 . . . . . 6 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
109ensymd 7952 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11 sdomentr 8039 . . . . 5 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
1210, 11sylan2 491 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13 alephon 8837 . . . . . 6 (ℵ‘𝐴) ∈ On
14 cardon 8715 . . . . . . 7 (card‘𝐵) ∈ On
15 onenon 8720 . . . . . . 7 ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
1614, 15ax-mp 5 . . . . . 6 (card‘𝐵) ∈ dom card
17 cardsdomel 8745 . . . . . 6 (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
1813, 16, 17mp2an 707 . . . . 5 ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
191eleq2i 2696 . . . . 5 ((ℵ‘𝐴) ∈ (card‘(card‘𝐵)) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
2018, 19bitri 264 . . . 4 ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
2112, 20sylib 208 . . 3 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ∈ (card‘𝐵))
22 ensdomtr 8041 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
239, 22mpancom 702 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
2423adantl 482 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
25 onenon 8720 . . . . . . 7 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
264, 25ax-mp 5 . . . . . 6 (ℵ‘suc 𝐴) ∈ dom card
27 cardsdomel 8745 . . . . . 6 (((card‘𝐵) ∈ On ∧ (ℵ‘suc 𝐴) ∈ dom card) → ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴))))
2814, 26, 27mp2an 707 . . . . 5 ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29 alephcard 8838 . . . . . 6 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
3029eleq2i 2696 . . . . 5 ((card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3128, 30bitri 264 . . . 4 ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3224, 31sylib 208 . . 3 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ∈ (ℵ‘suc 𝐴))
3321, 32jca 554 . 2 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
343, 33mto 188 1 ¬ ((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992   class class class wbr 4618  dom cdm 5079  Oncon0 5685  suc csuc 5687  ‘cfv 5850   ≈ cen 7897   ≼ cdom 7898   ≺ csdm 7899  cardccrd 8706  ℵcale 8707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-har 8408  df-card 8710  df-aleph 8711 This theorem is referenced by:  alephsucdom  8847  alephsucpw2  8879  alephgch  9441  winalim2  9463  aleph1re  14894
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