Theorem alephnbtwn 9983

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 9981	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ On
2		id 22	. . . . . . . . . 10 ⊢ ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3		cardon 9858	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
4	2, 3	eqeltrrdi 2844	. . . . . . . . 9 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ On)
5		onenon 9863	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ dom card)
7		cardsdomel 9888	. . . . . . . 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 2824	. . . . . . 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 9980	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13		onenon 9863	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14		harval2 9911	. . . . . . . . . . . 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 2786	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
17	16	eleq2d 2821	. . . . . . . . 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 5101	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
20	19	onnminsb 7744	. . . . . . . 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 4291	. . . . . . 7 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29		alephfnon 9977	. . . . . . . . . 10 ⊢ ℵ Fn On
30	29	fndmi 6595	. . . . . . . . 9 ⊢ dom ℵ = On
31	30	eleq2i 2827	. . . . . . . 8 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
32		ndmfv 6865	. . . . . . . 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 7759	. . . . . 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 3398  ∅c0 4284  ∩ cint 4901   class class class wbr 5097  dom cdm 5623  Oncon0 6316  suc csuc 6318  ‘cfv 6491   ≺ csdm 8884  harchar 9463  cardccrd 9849  ℵcale 9850

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-oi 9417  df-har 9464  df-card 9853  df-aleph 9854

This theorem is referenced by:  alephnbtwn2  9984