Theorem alephnbtwn 7908

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 7906	. . . . . . . 8 |- ( aleph ` A ) e. On
2		id 20	. . . . . . . . . 10 |- ( ( card ` B ) = B -> ( card ` B ) = B )
3		cardon 7787	. . . . . . . . . 10 |- ( card ` B ) e. On
4	2, 3	syl6eqelr 2493	. . . . . . . . 9 |- ( ( card ` B ) = B -> B e. On )
5		onenon 7792	. . . . . . . . 9 |- ( B e. On -> B e. dom card )
6	4, 5	syl 16	. . . . . . . 8 |- ( ( card ` B ) = B -> B e. dom card )
7		cardsdomel 7817	. . . . . . . 8 |- ( ( ( aleph ` A ) e. On /\ B e. dom card ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
8	1, 6, 7	sylancr 645	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. ( card ` B ) ) )
9		eleq2 2465	. . . . . . 7 |- ( ( card ` B ) = B -> ( ( aleph ` A ) e. ( card ` B ) <-> ( aleph ` A ) e. B ) )
10	8, 9	bitrd 245	. . . . . 6 |- ( ( card ` B ) = B -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
11	10	adantl 453	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B <-> ( aleph ` A ) e. B ) )
12		alephsuc 7905	. . . . . . . . . . 11 |- ( A e. On -> ( aleph ` suc A ) = (har ` ( aleph ` A ) ) )
13		onenon 7792	. . . . . . . . . . . 12 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
14		harval2 7840	. . . . . . . . . . . 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 2452	. . . . . . . . . 10 |- ( A e. On -> ( aleph ` suc A ) = |^| { x e. On | ( aleph ` A ) ~< x } )
17	16	eleq2d 2471	. . . . . . . . 9 |- ( A e. On -> ( B e. ( aleph ` suc A ) <-> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
18	17	biimpd 199	. . . . . . . 8 |- ( A e. On -> ( B e. ( aleph ` suc A ) -> B e. |^| { x e. On | ( aleph ` A ) ~< x } ) )
19		breq2 4176	. . . . . . . . 9 |- ( x = B -> ( ( aleph ` A ) ~< x <-> ( aleph ` A ) ~< B ) )
20	19	onnminsb 4743	. . . . . . . 8 |- ( B e. On -> ( B e. |^| { x e. On | ( aleph ` A ) ~< x } -> -. ( aleph ` A ) ~< B ) )
21	18, 20	sylan9 639	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< B ) )
22	21	con2d 109	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
23	4, 22	sylan2 461	. . . . 5 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) ~< B -> -. B e. ( aleph ` suc A ) ) )
24	11, 23	sylbird 227	. . . 4 |- ( ( A e. On /\ ( card ` B ) = B ) -> ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) )
25		imnan 412	. . . 4 |- ( ( ( aleph ` A ) e. B -> -. B e. ( aleph ` suc A ) ) <-> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
26	24, 25	sylib 189	. . 3 |- ( ( A e. On /\ ( card ` B ) = B ) -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
27	26	ex 424	. 2 |- ( A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
28		n0i 3593	. . . . . . 7 |- ( B e. ( aleph ` suc A ) -> -. ( aleph ` suc A ) = (/) )
29		alephfnon 7902	. . . . . . . . . 10 |- aleph Fn On
30		fndm 5503	. . . . . . . . . 10 |- ( aleph Fn On -> dom aleph = On )
31	29, 30	ax-mp 8	. . . . . . . . 9 |- dom aleph = On
32	31	eleq2i 2468	. . . . . . . 8 |- ( suc A e. dom aleph <-> suc A e. On )
33		ndmfv 5714	. . . . . . . 8 |- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) )
34	32, 33	sylnbir 299	. . . . . . 7 |- ( -. suc A e. On -> ( aleph ` suc A ) = (/) )
35	28, 34	nsyl2 121	. . . . . 6 |- ( B e. ( aleph ` suc A ) -> suc A e. On )
36		sucelon 4756	. . . . . 6 |- ( A e. On <-> suc A e. On )
37	35, 36	sylibr 204	. . . . 5 |- ( B e. ( aleph ` suc A ) -> A e. On )
38	37	adantl 453	. . . 4 |- ( ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) -> A e. On )
39	38	con3i 129	. . 3 |- ( -. A e. On -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )
40	39	a1d 23	. 2 |- ( -. A e. On -> ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) ) )
41	27, 40	pm2.61i 158	1 |- ( ( card ` B ) = B -> -. ( ( aleph ` A ) e. B /\ B e. ( aleph ` suc A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   {crab 2670   (/)c0 3588   |^|cint 4010   class class class wbr 4172   Oncon0 4541   suc csuc 4543   dom cdm 4837   Fn wfn 5408   ` cfv 5413   ~< csdm 7067  harchar 7480   cardccrd 7778   alephcale 7779

This theorem is referenced by:  alephnbtwn2  7909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783