Theorem alephnbtwn 8894

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 8892	. . . . . . . 8 |- ( aleph ` A ) e. On
2		id 22	. . . . . . . . . 10 |- ( ( card ` B ) = B -> ( card ` B ) = B )
3		cardon 8770	. . . . . . . . . 10 |- ( card ` B ) e. On
4	2, 3	syl6eqelr 2710	. . . . . . . . 9 |- ( ( card ` B ) = B -> B e. On )
5		onenon 8775	. . . . . . . . 9 |- ( B e. On -> B e. dom card )
6	4, 5	syl 17	. . . . . . . 8 |- ( ( card ` B ) = B -> B e. dom card )
7		cardsdomel 8800	. . . . . . . 8 |- ( ( ( aleph ` A ) e. On /\ B e. dom card ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
8	1, 6, 7	sylancr 695	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
9		eleq2 2690	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) e. ( card ` B ) <-> ( aleph ` A ) e. B ) )
10	8, 9	bitrd 268	. . . . . 6 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
11	10	adantl 482	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
12		alephsuc 8891	. . . . . . . . . . 11 |- ( A e. On -> ( aleph ` suc A ) = (har ` ( aleph ` A ) ) )
13		onenon 8775	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
14		harval2 8823	. . . . . . . . . . . 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 2672	. . . . . . . . . 10 |- ( A e. On -> ( aleph ` suc A ) = |^| { x e. On | ( aleph ` A ) ~< x } )
17	16	eleq2d 2687	. . . . . . . . 9 |- ( A e. On -> ( B e. ( aleph ` suc A ) <-> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
18	17	biimpd 219	. . . . . . . 8 |- ( A e. On -> ( B e. ( aleph ` suc A ) -> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
19		breq2 4657	. . . . . . . . 9 |- ( x = B -> ( ( aleph ` A ) ~< x <-> ( aleph ` A ) ~< B ) )
20	19	onnminsb 7004	. . . . . . . 8 |- ( B e. On -> ( B e. |^| { x e. On | ( aleph ` A ) ~< x } -> -. ( aleph ` A ) ~< B ) )
21	18, 20	sylan9 689	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< B ) )
22	21	con2d 129	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
23	4, 22	sylan2 491	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
24	11, 23	sylbird 250	. . . 4 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) )
25		imnan 438	. . . 4 |- ( ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) <-> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
26	24, 25	sylib 208	. . 3 |- ( ( A e. On /\ ( card ` B ) = B ) -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
27	26	ex 450	. 2 |- ( A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
28		n0i 3920	. . . . . . 7 |- ( B e. ( aleph ` suc A ) -> -. ( aleph ` suc A ) = (/) )
29		alephfnon 8888	. . . . . . . . . 10 |- aleph Fn On
30		fndm 5990	. . . . . . . . . 10 |- ( aleph Fn On -> dom aleph = On )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- dom aleph = On
32	31	eleq2i 2693	. . . . . . . 8 |- ( suc A e. dom aleph <-> suc A e. On )
33		ndmfv 6218	. . . . . . . 8 |- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) )
34	32, 33	sylnbir 321	. . . . . . 7 |- ( -. suc A e. On -> ( aleph ` suc A ) = (/) )
35	28, 34	nsyl2 142	. . . . . 6 |- ( B e. ( aleph ` suc A ) -> suc A e. On )
36		sucelon 7017	. . . . . 6 |- ( A e. On <-> suc A e. On )
37	35, 36	sylibr 224	. . . . 5 |- ( B e. ( aleph ` suc A ) -> A e. On )
38	37	adantl 482	. . . 4 |- ( ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) -> A e. On )
39	38	con3i 150	. . 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 176	1 |- ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   (/)c0 3915   |^|cint 4475   class class class wbr 4653   dom cdm 5114   Oncon0 5723   suc csuc 5725   Fn wfn 5883   ` cfv 5888   ~< csdm 7954  harchar 8461   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephnbtwn2  8895