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Theorem alephnbtwn2 8933
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 8823 . . 3 (card‘(card‘𝐵)) = (card‘𝐵)
2 alephnbtwn 8932 . . 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 8930 . . . . . . . 8 (ℵ‘suc 𝐴) ∈ On
5 sdomdom 8025 . . . . . . . 8 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6 ondomen 8898 . . . . . . . 8 (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
74, 5, 6sylancr 696 . . . . . . 7 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8 cardid2 8817 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
97, 8syl 17 . . . . . 6 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
109ensymd 8048 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11 sdomentr 8135 . . . . 5 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
1210, 11sylan2 490 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13 alephon 8930 . . . . . 6 (ℵ‘𝐴) ∈ On
14 cardon 8808 . . . . . . 7 (card‘𝐵) ∈ On
15 onenon 8813 . . . . . . 7 ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
1614, 15ax-mp 5 . . . . . 6 (card‘𝐵) ∈ dom card
17 cardsdomel 8838 . . . . . 6 (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
1813, 16, 17mp2an 708 . . . . 5 ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
191eleq2i 2722 . . . . 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 8137 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
239, 22mpancom 704 . . . . 5 (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
2423adantl 481 . . . 4 (((ℵ‘𝐴) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
25 onenon 8813 . . . . . . 7 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
264, 25ax-mp 5 . . . . . 6 (ℵ‘suc 𝐴) ∈ dom card
27 cardsdomel 8838 . . . . . 6 (((card‘𝐵) ∈ On ∧ (ℵ‘suc 𝐴) ∈ dom card) → ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴))))
2814, 26, 27mp2an 708 . . . . 5 ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29 alephcard 8931 . . . . . 6 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
3029eleq2i 2722 . . . . 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 553 . 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 383   = wceq 1523  wcel 2030   class class class wbr 4685  dom cdm 5143  Oncon0 5761  suc csuc 5763  cfv 5926  cen 7994  cdom 7995  csdm 7996  cardccrd 8799  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804
This theorem is referenced by:  alephsucdom  8940  alephsucpw2  8972  alephgch  9534  winalim2  9556  aleph1re  15018
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