Theorem alephnbtwn 9995

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 9993	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ On
2		id 22	. . . . . . . . . 10 ⊢ ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3		cardon 9870	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
4	2, 3	eqeltrrdi 2846	. . . . . . . . 9 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ On)
5		onenon 9875	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ dom card)
7		cardsdomel 9900	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ∈ On ∧ 𝐵 ∈ dom card) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
8	1, 6, 7	sylancr 588	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ (card‘𝐵)))
9		eleq2 2826	. . . . . . 7 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ∈ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ 𝐵))
10	8, 9	bitrd 279	. . . . . 6 ⊢ ((card‘𝐵) = 𝐵 → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
11	10	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12		alephsuc 9992	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13		onenon 9875	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14		harval2 9923	. . . . . . . . . . . 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 2788	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
17	16	eleq2d 2823	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
18	17	biimpd 229	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥}))
19		breq2 5104	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
20	19	onnminsb 7756	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
21	18, 20	sylan9 507	. . . . . . 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 594	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
24	11, 23	sylbird 260	. . . 4 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)))
25		imnan 399	. . . 4 ⊢ (((ℵ‘𝐴) ∈ 𝐵 → ¬ 𝐵 ∈ (ℵ‘suc 𝐴)) ↔ ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
26	24, 25	sylib 218	. . 3 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴)))
27	26	ex 412	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
28		n0i 4294	. . . . . . 7 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29		alephfnon 9989	. . . . . . . . . 10 ⊢ ℵ Fn On
30	29	fndmi 6606	. . . . . . . . 9 ⊢ dom ℵ = On
31	30	eleq2i 2829	. . . . . . . 8 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
32		ndmfv 6876	. . . . . . . 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		onsucb 7771	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
36	34, 35	sylibr 234	. . . . 5 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
37	36	adantl 481	. . . 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  ∅c0 4287  ∩ cint 4904   class class class wbr 5100  dom cdm 5634  Oncon0 6327  suc csuc 6329  ‘cfv 6502   ≺ csdm 8896  harchar 9475  cardccrd 9861  ℵcale 9862

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-oi 9429  df-har 9476  df-card 9865  df-aleph 9866

This theorem is referenced by:  alephnbtwn2  9996