Theorem alephnbtwn2 9984

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 9873	. . 3 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
2		alephnbtwn 9983	. . 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 9981	. . . . . . . 8 ⊢ (ℵ‘suc 𝐴) ∈ On
5		sdomdom 8919	. . . . . . . 8 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≼ (ℵ‘suc 𝐴))
6		ondomen 9949	. . . . . . . 8 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ 𝐵 ≼ (ℵ‘suc 𝐴)) → 𝐵 ∈ dom card)
7	4, 5, 6	sylancr 588	. . . . . . 7 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ∈ dom card)
8		cardid2 9867	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → (card‘𝐵) ≈ 𝐵)
10	9	ensymd 8944	. . . . 5 ⊢ (𝐵 ≺ (ℵ‘suc 𝐴) → 𝐵 ≈ (card‘𝐵))
11		sdomentr 9041	. . . . 5 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → (ℵ‘𝐴) ≺ (card‘𝐵))
12	10, 11	sylan2 594	. . . 4 ⊢ (((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝐴) ≺ (card‘𝐵))
13		alephon 9981	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
14		cardon 9858	. . . . . . 7 ⊢ (card‘𝐵) ∈ On
15		onenon 9863	. . . . . . 7 ⊢ ((card‘𝐵) ∈ On → (card‘𝐵) ∈ dom card)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (card‘𝐵) ∈ dom card
17		cardsdomel 9888	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ On ∧ (card‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵))))
18	13, 16, 17	mp2an 693	. . . . 5 ⊢ ((ℵ‘𝐴) ≺ (card‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(card‘𝐵)))
19	1	eleq2i 2827	. . . . 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 9043	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) → (card‘𝐵) ≺ (ℵ‘suc 𝐴))
23	9, 22	mpancom 689	. . . . 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 9863	. . . . . . 7 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
26	4, 25	ax-mp 5	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ dom card
27		cardsdomel 9888	. . . . . 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 693	. . . . 5 ⊢ ((card‘𝐵) ≺ (ℵ‘suc 𝐴) ↔ (card‘𝐵) ∈ (card‘(ℵ‘suc 𝐴)))
29		alephcard 9982	. . . . . 6 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
30	29	eleq2i 2827	. . . . 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 1542   ∈ wcel 2114   class class class wbr 5097  dom cdm 5623  Oncon0 6316  suc csuc 6318  ‘cfv 6491   ≈ cen 8882   ≼ cdom 8883   ≺ csdm 8884  cardccrd 9849  ℵcale 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-oi 9417  df-har 9464  df-card 9853  df-aleph 9854

This theorem is referenced by:  alephsucdom  9991  alephsucpw2  10023  alephgch  10587  winalim2  10609  aleph1re  16172