Theorem alephnbtwn2 4841

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		cardidm 4821	. . . 4 ⊢ (card ‘(card ‘B)) = (card ‘B)
2		alephnbtwn 4840	. . . 4 ⊢ ((card ‘(card ‘B)) = (card ‘B) → ¬ ((ℵ ‘A) ∈ (card ‘B) ⋀ (card ‘B) ∈ (ℵ ‘suc A)))
3	1, 2	ax-mp 7	. . 3 ⊢ ¬ ((ℵ ‘A) ∈ (card ‘B) ⋀ (card ‘B) ∈ (ℵ ‘suc A))
4		alephon 4837	. . . . . 6 ⊢ (ℵ ‘A) ∈ On
5		cardsdomel 4824	. . . . . 6 ⊢ ((ℵ ‘A) ∈ On → ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B)))
6	4, 5	ax-mp 7	. . . . 5 ⊢ ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B))
7	6	a1i 8	. . . 4 ⊢ (B ∈ V → ((ℵ ‘A) ≺ B ↔ (ℵ ‘A) ∈ (card ‘B)))
8		alephon 4837	. . . . . 6 ⊢ (ℵ ‘suc A) ∈ On
9		cardsdom 4809	. . . . . 6 ⊢ ((B ∈ V ⋀ (ℵ ‘suc A) ∈ On) → ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ B ≺ (ℵ ‘suc A)))
10	8, 9	mpan2 694	. . . . 5 ⊢ (B ∈ V → ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ B ≺ (ℵ ‘suc A)))
11		alephcard 4839	. . . . . 6 ⊢ (card ‘(ℵ ‘suc A)) = (ℵ ‘suc A)
12	11	eleq2i 1530	. . . . 5 ⊢ ((card ‘B) ∈ (card ‘(ℵ ‘suc A)) ↔ (card ‘B) ∈ (ℵ ‘suc A))
13	10, 12	syl5rbbr 533	. . . 4 ⊢ (B ∈ V → (B ≺ (ℵ ‘suc A) ↔ (card ‘B) ∈ (ℵ ‘suc A)))
14	7, 13	anbi12d 626	. . 3 ⊢ (B ∈ V → (((ℵ ‘A) ≺ B ⋀ B ≺ (ℵ ‘suc A)) ↔ ((ℵ ‘A) ∈ (card ‘B) ⋀ (card ‘B) ∈ (ℵ ‘suc A))))
15	3, 14	mtbiri 715	. 2 ⊢ (B ∈ V → ¬ ((ℵ ‘A) ≺ B ⋀ B ≺ (ℵ ‘suc A)))
16		relsdom 4356	. . . . 5 ⊢ Rel ≺
17	16	brrelexi 3198	. . . 4 ⊢ (B ≺ (ℵ ‘suc A) → B ∈ V)
18	17	adantl 388	. . 3 ⊢ (((ℵ ‘A) ≺ B ⋀ B ≺ (ℵ ‘suc A)) → B ∈ V)
19	18	con3i 98	. 2 ⊢ (¬ B ∈ V → ¬ ((ℵ ‘A) ≺ B ⋀ B ≺ (ℵ ‘suc A)))
20	15, 19	pm2.61i 126	1 ⊢ ¬ ((ℵ ‘A) ≺ B ⋀ B ≺ (ℵ ‘suc A))

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  Vcvv 1802   class class class wbr 2609  Oncon0 2938  suc csuc 2940   ‘cfv 3172   ≺ csdm 4350  cardccrd 4785  ℵcale 4786

This theorem is referenced by:  alephsucpw 4842  alephsucdom 4852  aleph1re 7494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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