Theorem alephnbtwn 8469

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 8467	. . . . . . . 8 |- ( aleph ` A ) e. On
2		id 22	. . . . . . . . . 10 |- ( ( card ` B ) = B -> ( card ` B ) = B )
3		cardon 8342	. . . . . . . . . 10 |- ( card ` B ) e. On
4	2, 3	syl6eqelr 2554	. . . . . . . . 9 |- ( ( card ` B ) = B -> B e. On )
5		onenon 8347	. . . . . . . . 9 |- ( B e. On -> B e. dom card )
6	4, 5	syl 16	. . . . . . . 8 |- ( ( card ` B ) = B -> B e. dom card )
7		cardsdomel 8372	. . . . . . . 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 2530	. . . . . . 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 8466	. . . . . . . . . . 11 |- ( A e. On -> ( aleph ` suc A ) = (har ` ( aleph ` A ) ) )
13		onenon 8347	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
14		harval2 8395	. . . . . . . . . . . 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 2514	. . . . . . . . . 10 |- ( A e. On -> ( aleph ` suc A ) = |^| { x e. On | ( aleph ` A ) ~< x } )
17	16	eleq2d 2527	. . . . . . . . 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 4460	. . . . . . . . 9 |- ( x = B -> ( ( aleph ` A ) ~< x <-> ( aleph ` A ) ~< B ) )
20	19	onnminsb 6638	. . . . . . . 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 3798	. . . . . . 7 |- ( B e. ( aleph ` suc A ) -> -. ( aleph ` suc A ) = (/) )
29		alephfnon 8463	. . . . . . . . . 10 |- aleph Fn On
30		fndm 5686	. . . . . . . . . 10 |- ( aleph Fn On -> dom aleph = On )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- dom aleph = On
32	31	eleq2i 2535	. . . . . . . 8 |- ( suc A e. dom aleph <-> suc A e. On )
33		ndmfv 5896	. . . . . . . 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 6651	. . . . . 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 1395   e. wcel 1819   {crab 2811   (/)c0 3793   |^|cint 4288   class class class wbr 4456   Oncon0 4887   suc csuc 4889   dom cdm 5008   Fn wfn 5589   ` cfv 5594   ~< csdm 7534  harchar 8000   cardccrd 8333   alephcale 8334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338

This theorem is referenced by:  alephnbtwn2  8470