Theorem alephnbtwn 10053

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 10051	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ On
2		id 22	. . . . . . . . . 10 ⊢ ((card‘𝐵) = 𝐵 → (card‘𝐵) = 𝐵)
3		cardon 9926	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
4	2, 3	eqeltrrdi 2843	. . . . . . . . 9 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ On)
5		onenon 9931	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((card‘𝐵) = 𝐵 → 𝐵 ∈ dom card)
7		cardsdomel 9956	. . . . . . . 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 2823	. . . . . . 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 483	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (card‘𝐵) = 𝐵) → ((ℵ‘𝐴) ≺ 𝐵 ↔ (ℵ‘𝐴) ∈ 𝐵))
12		alephsuc 10050	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
13		onenon 9931	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
14		harval2 9979	. . . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥})
17	16	eleq2d 2820	. . . . . . . . 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 5148	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ 𝑥 ↔ (ℵ‘𝐴) ≺ 𝐵))
20	19	onnminsb 7774	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ {𝑥 ∈ On ∣ (ℵ‘𝐴) ≺ 𝑥} → ¬ (ℵ‘𝐴) ≺ 𝐵))
21	18, 20	sylan9 509	. . . . . . 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 401	. . . 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 414	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))))
28		n0i 4331	. . . . . . 7 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘suc 𝐴) = ∅)
29		alephfnon 10047	. . . . . . . . . 10 ⊢ ℵ Fn On
30	29	fndmi 6645	. . . . . . . . 9 ⊢ dom ℵ = On
31	30	eleq2i 2826	. . . . . . . 8 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
32		ndmfv 6916	. . . . . . . 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 7792	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
36	34, 35	sylibr 233	. . . . 5 ⊢ (𝐵 ∈ (ℵ‘suc 𝐴) → 𝐴 ∈ On)
37	36	adantl 483	. . . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  ∅c0 4320  ∩ cint 4946   class class class wbr 5144  dom cdm 5672  Oncon0 6356  suc csuc 6358  ‘cfv 6535   ≺ csdm 8926  harchar 9538  cardccrd 9917  ℵcale 9918

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-inf2 9623

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-om 7843  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-er 8691  df-en 8928  df-dom 8929  df-sdom 8930  df-oi 9492  df-har 9539  df-card 9921  df-aleph 9922

This theorem is referenced by:  alephnbtwn2  10054