Theorem alephnbtwn 9491

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) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))

Proof of Theorem alephnbtwn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephon 9489	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ On
2		id 22	. . . . . . . . . 10 ⊢ ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3		cardon 9367	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
4	2, 3	eqeltrrdi 2922	. . . . . . . . 9 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ On)
5		onenon 9372	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ dom card)
7		cardsdomel 9397	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
8	1, 6, 7	sylancr 589	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9		eleq2 2901	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
10	8, 9	bitrd 281	. . . . . 6 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
11	10	adantl 484	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12		alephsuc 9488	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13		onenon 9372	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14		harval2 9420	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ dom card → (har‘(ℵ‘𝐴)) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
15	1, 13, 14	mp2b 10	. . . . . . . . . . 11 ⊢ (har‘(ℵ‘𝐴)) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}
16	12, 15	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
17	16	eleq2d 2898	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
18	17	biimpd 231	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19		breq2 5062	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
20	19	onnminsb 7513	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
21	18, 20	sylan9 510	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
22	21	con2d 136	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
23	4, 22	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
24	11, 23	sylbird 262	. . . 4 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25		imnan 402	. . . 4 ⊢ (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
26	24, 25	sylib 220	. . 3 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
27	26	ex 415	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
28		n0i 4298	. . . . . . 7 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29		alephfnon 9485	. . . . . . . . . 10 ⊢ ℵ Fn On
30		fndm 6449	. . . . . . . . . 10 ⊢ (ℵ Fn On → dom ℵ = On)
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ dom ℵ = On
32	31	eleq2i 2904	. . . . . . . 8 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
33		ndmfv 6694	. . . . . . . 8 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
34	32, 33	sylnbir 333	. . . . . . 7 ⊢ (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
35	28, 34	nsyl2 143	. . . . . 6 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
36		sucelon 7526	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
37	35, 36	sylibr 236	. . . . 5 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
38	37	adantl 484	. . . 4 ⊢ (((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
39	38	con3i 157	. . 3 ⊢ (¬ 𝐴 ∈ On → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
40	39	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
41	27, 40	pm2.61i 184	1 ⊢ ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142  ∅c0 4290  ∩ cint 4868   class class class wbr 5058  dom cdm 5549  Oncon0 6185  suc csuc 6187   Fn wfn 6344  ‘cfv 6349   ≺ csdm 8502  harchar 9014  cardccrd 9358  ℵcale 9359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-oi 8968  df-har 9016  df-card 9362  df-aleph 9363

This theorem is referenced by:  alephnbtwn2  9492