Theorem alephnbtwn2 3675

Description: No set has equinumerosity between an aleph and its successor aleph.

Assertion

Ref	Expression
alephnbtwn2	⊢ ¬ ((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A))

Proof of Theorem alephnbtwn2

Step	Hyp	Ref	Expression
1		cardcard 3655	. . . 4 ⊢ (card ‘(card ‘B)) = (card ‘B)
2		alephnbtwn 3674	. . . 4 ⊢ ((card ‘(card ‘B)) = (card ‘B) → ¬ ((ℵ ‘A) ∈ (card ‘B) ∧ (card ‘B) ∈ (ℵ ‘suc A)))
3	1, 2	ax-mp 6	. . 3 ⊢ ¬ ((ℵ ‘A) ∈ (card ‘B) ∧ (card ‘B) ∈ (ℵ ‘suc A))
4		alephon 3671	. . . . . 6 ⊢ (ℵ ‘A) ∈ On
5		cardsdomel 3658	. . . . . 6 ⊢ ((ℵ ‘A) ∈ On → ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B)))
6	4, 5	ax-mp 6	. . . . 5 ⊢ ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B))
7	6	a1i 7	. . . 4 ⊢ (B ∈ V → ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B)))
8		alephon 3671	. . . . . 6 ⊢ (ℵ ‘suc A) ∈ On
9		cardsdom 3643	. . . . . 6 ⊢ ((B ∈ V ∧ (ℵ ‘suc A) ∈ On) → ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ B ≺ (ℵ ‘suc A)))
10	8, 9	mpan2 519	. . . . 5 ⊢ (B ∈ V → ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ B ≺ (ℵ ‘suc A)))
11		alephcard 3673	. . . . . 6 ⊢ (card ‘(ℵ ‘suc A)) = (ℵ ‘suc A)
12	11	eleq2i 1153	. . . . 5 ⊢ ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ (card ‘B) ∈ (ℵ ‘suc A))
13	10, 12	syl5rbbr 413	. . . 4 ⊢ (B ∈ V → (B ≺ (ℵ ‘suc A) ↔ (card ‘B) ∈ (ℵ ‘suc A)))
14	7, 13	anbi12d 476	. . 3 ⊢ (B ∈ V → (((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A)) ↔ ((ℵ ‘A) ∈ (card ‘B) ∧ (card ‘B) ∈ (ℵ ‘suc A))))
15	3, 14	mtbiri 539	. 2 ⊢ (B ∈ V → ¬ ((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A)))
16		relsdom 3279	. . . . 5 ⊢ Rel ≺
17	16	brrelexi 2447	. . . 4 ⊢ (B ≺ (ℵ ‘suc A) → B ∈ V)
18	17	adantl 305	. . 3 ⊢ (((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A)) → B ∈ V)
19	18	con3i 90	. 2 ⊢ (¬ B ∈ V → ¬ ((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A)))
20	15, 19	pm2.61i 110	1 ⊢ ¬ ((ℵ ‘A) ≺ B ∧ B ≺ (ℵ ‘suc A))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  Oncon0 2199  suc csuc 2201   ‘cfv 2422   ≺ csdm 3273  cardccrd 3620  ℵcale 3621

This theorem is referenced by:  aleph1 3676  alephsucdom 3685

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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