Theorem alephnbtwn2 9966

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 9855	. . 3 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
2		alephnbtwn 9965	. . 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 9963	. . . . . . . 8 ⊢ (ℵ‘suc 𝐴) ∈ On
5		sdomdom 8905	. . . . . . . 8 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6		ondomen 9931	. . . . . . . 8 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
7	4, 5, 6	sylancr 587	. . . . . . 7 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8		cardid2 9849	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
10	9	ensymd 8930	. . . . 5 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11		sdomentr 9028	. . . . 5 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
12	10, 11	sylan2 593	. . . 4 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13		alephon 9963	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
14		cardon 9840	. . . . . . 7 ⊢ (card‘𝐵) ∈ On
15		onenon 9845	. . . . . . 7 ⊢ ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (card‘𝐵) ∈ dom card
17		cardsdomel 9870	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
18	13, 16, 17	mp2an 692	. . . . 5 ⊢ ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
19	1	eleq2i 2820	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ (card‘(card‘𝐵)) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
20	18, 19	bitri 275	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘𝐵))
21	12, 20	sylib 218	. . 3 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ∈ (card‘𝐵))
22		ensdomtr 9030	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
23	9, 22	mpancom 688	. . . . 5 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
24	23	adantl 481	. . . 4 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
25		onenon 9845	. . . . . . 7 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
26	4, 25	ax-mp 5	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ dom card
27		cardsdomel 9870	. . . . . 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 692	. . . . 5 ⊢ ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29		alephcard 9964	. . . . . 6 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
30	29	eleq2i 2820	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
31	28, 30	bitri 275	. . . 4 ⊢ ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (ℵ‘suc 𝐴))
32	24, 31	sylib 218	. . 3 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ∈ (ℵ‘suc 𝐴))
33	21, 32	jca 511	. 2 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → ((ℵ‘𝐴) ∈ (card‘𝐵) ∧ (card‘𝐵) ∈ (ℵ‘suc 𝐴)))
34	3, 33	mto 197	1 ⊢ ¬ ((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5092  dom cdm 5619  Oncon0 6307  suc csuc 6309  ‘cfv 6482   ≈ cen 8869   ≼ cdom 8870   ≺ csdm 8871  cardccrd 9831  ℵcale 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-oi 9402  df-har 9449  df-card 9835  df-aleph 9836

This theorem is referenced by:  alephsucdom  9973  alephsucpw2  10005  alephgch  10568  winalim2  10590  aleph1re  16154