Theorem alephnbtwn 9827

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 9825	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ On
2		id 22	. . . . . . . . . 10 ⊢ ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3		cardon 9702	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
4	2, 3	eqeltrrdi 2848	. . . . . . . . 9 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ On)
5		onenon 9707	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ dom card)
7		cardsdomel 9732	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
8	1, 6, 7	sylancr 587	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9		eleq2 2827	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
10	8, 9	bitrd 278	. . . . . 6 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
11	10	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12		alephsuc 9824	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13		onenon 9707	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14		harval2 9755	. . . . . . . . . . . 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	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
17	16	eleq2d 2824	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
18	17	biimpd 228	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19		breq2 5078	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
20	19	onnminsb 7649	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
21	18, 20	sylan9 508	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ 𝐵))
22	21	con2d 134	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
23	4, 22	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
24	11, 23	sylbird 259	. . . 4 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25		imnan 400	. . . 4 ⊢ (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
26	24, 25	sylib 217	. . 3 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
27	26	ex 413	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
28		n0i 4267	. . . . . . 7 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29		alephfnon 9821	. . . . . . . . . 10 ⊢ ℵ Fn On
30	29	fndmi 6537	. . . . . . . . 9 ⊢ dom ℵ = On
31	30	eleq2i 2830	. . . . . . . 8 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
32		ndmfv 6804	. . . . . . . 8 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
33	31, 32	sylnbir 331	. . . . . . 7 ⊢ (¬ suc 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
34	28, 33	nsyl2 141	. . . . . 6 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → suc 𝐴 ∈ On)
35		sucelon 7664	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
36	34, 35	sylibr 233	. . . . 5 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
37	36	adantl 482	. . . 4 ⊢ (((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)) → 𝐴 ∈ On)
38	37	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ On → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
39	38	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
40	27, 39	pm2.61i 182	1 ⊢ ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  ∅c0 4256  ∩ cint 4879   class class class wbr 5074  dom cdm 5589  Oncon0 6266  suc csuc 6268  ‘cfv 6433   ≺ csdm 8732  harchar 9315  cardccrd 9693  ℵcale 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698

This theorem is referenced by:  alephnbtwn2  9828