Theorem alephnbtwn2 9989

Description: No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

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

Proof of Theorem alephnbtwn2

Step	Hyp	Ref	Expression
1		cardidm 9878	. . 3 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
2		alephnbtwn 9988	. . 3 ⊢ ((card‘(card‘𝐵)) = (card‘𝐵) → ¬ ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
3	1, 2	ax-mp 5	. 2 ⊢ ¬ ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
4		alephon 9986	. . . . . . . 8 ⊢ (ℵ‘suc 𝐴) ∈ On
5		sdomdom 8921	. . . . . . . 8 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6		ondomen 9954	. . . . . . . 8 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
7	4, 5, 6	sylancr 594	. . . . . . 7 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8		cardid2 9872	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
10	9	ensymd 8946	. . . . 5 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11		sdomentr 9043	. . . . 5 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
12	10, 11	sylan2 600	. . . 4 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13		alephon 9986	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
14		cardon 9863	. . . . . . 7 ⊢ (card‘𝐵) ∈ On
15		onenon 9868	. . . . . . 7 ⊢ ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (card‘𝐵) ∈ dom card
17		cardsdomel 9893	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
18	13, 16, 17	mp2an 699	. . . . 5 ⊢ ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
19	1	eleq2i 2833	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ (card‘(card‘𝐵)) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
20	18, 19	bitri 277	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
21	12, 20	sylib 220	. . 3 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ∈ (card‘𝐵))
22		ensdomtr 9045	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
23	9, 22	mpancom 695	. . . . 5 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
24	23	adantl 483	. . . 4 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
25		onenon 9868	. . . . . . 7 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
26	4, 25	ax-mp 5	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ dom card
27		cardsdomel 9893	. . . . . 6 ⊢ (((card‘𝐵) ∈ On ∧ (ℵ‘suc 𝐴) ∈ dom card) → ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴))))
28	14, 26, 27	mp2an 699	. . . . 5 ⊢ ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29		alephcard 9987	. . . . . 6 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
30	29	eleq2i 2833	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
31	28, 30	bitri 277	. . . 4 ⊢ ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
32	24, 31	sylib 220	. . 3 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ∈ (ℵ‘suc 𝐴))
33	21, 32	jca 517	. 2 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
34	3, 33	mto 199	1 ⊢ ¬ ((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   class class class wbr 5074  dom cdm 5620  Oncon0 6313  suc csuc 6315  ‘cfv 6488   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  cardccrd 9854  ℵcale 9855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-har 9466  df-card 9858  df-aleph 9859

This theorem is referenced by:  alephsucdom  9996  alephsucpw2  10028  alephgch  10593  winalim2  10615  aleph1re  16207