Theorem alephnbtwn 8520

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 8518	. . . . . . . 8 |- ( aleph ` A ) e. On
2		id 22	. . . . . . . . . 10 |- ( ( card ` B ) = B -> ( card ` B ) = B )
3		cardon 8396	. . . . . . . . . 10 |- ( card ` B ) e. On
4	2, 3	syl6eqelr 2558	. . . . . . . . 9 |- ( ( card ` B ) = B -> B e. On )
5		onenon 8401	. . . . . . . . 9 |- ( B e. On -> B e. dom card )
6	4, 5	syl 17	. . . . . . . 8 |- ( ( card ` B ) = B -> B e. dom card )
7		cardsdomel 8426	. . . . . . . 8 |- ( ( ( aleph ` A ) e. On /\ B e. dom card ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
8	1, 6, 7	sylancr 676	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
9		eleq2 2538	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) e. ( card ` B ) <-> ( aleph ` A ) e. B ) )
10	8, 9	bitrd 261	. . . . . 6 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
11	10	adantl 473	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
12		alephsuc 8517	. . . . . . . . . . 11 |- ( A e. On -> ( aleph ` suc A ) = (har ` ( aleph ` A ) ) )
13		onenon 8401	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
14		harval2 8449	. . . . . . . . . . . 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 2521	. . . . . . . . . 10 |- ( A e. On -> ( aleph ` suc A ) = |^| { x e. On | ( aleph ` A ) ~< x } )
17	16	eleq2d 2534	. . . . . . . . 9 |- ( A e. On -> ( B e. ( aleph ` suc A ) <-> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
18	17	biimpd 212	. . . . . . . 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 6650	. . . . . . . 8 |- ( B e. On -> ( B e. |^| { x e. On | ( aleph ` A ) ~< x } -> -. ( aleph ` A ) ~< B ) )
21	18, 20	sylan9 669	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< B ) )
22	21	con2d 119	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
23	4, 22	sylan2 482	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
24	11, 23	sylbird 243	. . . 4 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) )
25		imnan 429	. . . 4 |- ( ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) <-> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
26	24, 25	sylib 201	. . 3 |- ( ( A e. On /\ ( card ` B ) = B ) -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
27	26	ex 441	. 2 |- ( A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
28		n0i 3727	. . . . . . 7 |- ( B e. ( aleph ` suc A ) -> -. ( aleph ` suc A ) = (/) )
29		alephfnon 8514	. . . . . . . . . 10 |- aleph Fn On
30		fndm 5685	. . . . . . . . . 10 |- ( aleph Fn On -> dom aleph = On )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- dom aleph = On
32	31	eleq2i 2541	. . . . . . . 8 |- ( suc A e. dom aleph <-> suc A e. On )
33		ndmfv 5903	. . . . . . . 8 |- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) )
34	32, 33	sylnbir 314	. . . . . . 7 |- ( -. suc A e. On -> ( aleph ` suc A ) = (/) )
35	28, 34	nsyl2 132	. . . . . 6 |- ( B e. ( aleph ` suc A ) -> suc A e. On )
36		sucelon 6663	. . . . . 6 |- ( A e. On <-> suc A e. On )
37	35, 36	sylibr 217	. . . . 5 |- ( B e. ( aleph ` suc A ) -> A e. On )
38	37	adantl 473	. . . 4 |- ( ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) -> A e. On )
39	38	con3i 142	. . 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 169	1 |- ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   {crab 2760   (/)c0 3722   |^|cint 4226   class class class wbr 4395   dom cdm 4839   Oncon0 5430   suc csuc 5432   Fn wfn 5584   ` cfv 5589   ~< csdm 7586  harchar 8089   cardccrd 8387   alephcale 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392

This theorem is referenced by:  alephnbtwn2  8521