Theorem alephnbtwn 8347

Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
alephnbtwn	|- ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )

Proof of Theorem alephnbtwn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephon 8345	. . . . . . . 8 |- ( aleph ` A ) e. On
2		id 22	. . . . . . . . . 10 |- ( ( card ` B ) = B -> ( card ` B ) = B )
3		cardon 8220	. . . . . . . . . 10 |- ( card ` B ) e. On
4	2, 3	syl6eqelr 2549	. . . . . . . . 9 |- ( ( card ` B ) = B -> B e. On )
5		onenon 8225	. . . . . . . . 9 |- ( B e. On -> B e. dom card )
6	4, 5	syl 16	. . . . . . . 8 |- ( ( card ` B ) = B -> B e. dom card )
7		cardsdomel 8250	. . . . . . . 8 |- ( ( ( aleph ` A ) e. On /\ B e. dom card ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
8	1, 6, 7	sylancr 663	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
9		eleq2 2525	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) e. ( card ` B ) <-> ( aleph ` A ) e. B ) )
10	8, 9	bitrd 253	. . . . . 6 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
11	10	adantl 466	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
12		alephsuc 8344	. . . . . . . . . . 11 |- ( A e. On -> ( aleph ` suc A ) = (har ` ( aleph ` A ) ) )
13		onenon 8225	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
14		harval2 8273	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. dom card -> (har ` ( aleph ` A ) ) = |^| { x e. On | ( aleph ` A ) ~< x } )
15	1, 13, 14	mp2b 10	. . . . . . . . . . 11 |- (har ` ( aleph ` A ) ) = |^| { x e. On | ( aleph ` A ) ~< x }
16	12, 15	syl6eq 2509	. . . . . . . . . 10 |- ( A e. On -> ( aleph ` suc A ) = |^| { x e. On | ( aleph ` A ) ~< x } )
17	16	eleq2d 2522	. . . . . . . . 9 |- ( A e. On -> ( B e. ( aleph ` suc A ) <-> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
18	17	biimpd 207	. . . . . . . 8 |- ( A e. On -> ( B e. ( aleph ` suc A ) -> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
19		breq2 4399	. . . . . . . . 9 |- ( x = B -> ( ( aleph ` A ) ~< x <-> ( aleph ` A ) ~< B ) )
20	19	onnminsb 6520	. . . . . . . 8 |- ( B e. On -> ( B e. |^| { x e. On | ( aleph ` A ) ~< x } -> -. ( aleph ` A ) ~< B ) )
21	18, 20	sylan9 657	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< B ) )
22	21	con2d 115	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
23	4, 22	sylan2 474	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
24	11, 23	sylbird 235	. . . 4 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) )
25		imnan 422	. . . 4 |- ( ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) <-> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
26	24, 25	sylib 196	. . 3 |- ( ( A e. On /\ ( card ` B ) = B ) -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
27	26	ex 434	. 2 |- ( A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
28		n0i 3745	. . . . . . 7 |- ( B e. ( aleph ` suc A ) -> -. ( aleph ` suc A ) = (/) )
29		alephfnon 8341	. . . . . . . . . 10 |- aleph Fn On
30		fndm 5613	. . . . . . . . . 10 |- ( aleph Fn On -> dom aleph = On )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- dom aleph = On
32	31	eleq2i 2530	. . . . . . . 8 |- ( suc A e. dom aleph <-> suc A e. On )
33		ndmfv 5818	. . . . . . . 8 |- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) )
34	32, 33	sylnbir 307	. . . . . . 7 |- ( -. suc A e. On -> ( aleph ` suc A ) = (/) )
35	28, 34	nsyl2 127	. . . . . 6 |- ( B e. ( aleph ` suc A ) -> suc A e. On )
36		sucelon 6533	. . . . . 6 |- ( A e. On <-> suc A e. On )
37	35, 36	sylibr 212	. . . . 5 |- ( B e. ( aleph ` suc A ) -> A e. On )
38	37	adantl 466	. . . 4 |- ( ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) -> A e. On )
39	38	con3i 135	. . 3 |- ( -. A e. On -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
40	39	a1d 25	. 2 |- ( -. A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
41	27, 40	pm2.61i 164	1 |- ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2800   (/)c0 3740   |^|cint 4231   class class class wbr 4395   Oncon0 4822   suc csuc 4824   dom cdm 4943   Fn wfn 5516   ` cfv 5521   ~< csdm 7414  harchar 7877   cardccrd 8211   alephcale 8212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-oi 7830  df-har 7879  df-card 8215  df-aleph 8216

This theorem is referenced by:  alephnbtwn2  8348