Theorem alephexp1 9401

Description: An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
alephexp1	|- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ( 2o ^m ( aleph ` B ) ) )

Proof of Theorem alephexp1

Step	Hyp	Ref	Expression
1		alephon 8892	. . . 4 |- ( aleph ` B ) e. On
2		onenon 8775	. . . 4 |- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card )
3	1, 2	mp1i 13	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( aleph ` B ) e. dom card )
4		fvex 6201	. . . 4 |- ( aleph ` B ) e. _V
5		simplr 792	. . . . 5 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> B e. On )
6		alephgeom 8905	. . . . 5 |- ( B e. On <-> om C_ ( aleph ` B ) )
7	5, 6	sylib 208	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> om C_ ( aleph ` B ) )
8		ssdomg 8001	. . . 4 |- ( ( aleph ` B ) e. _V -> ( om C_ ( aleph ` B ) -> om ~<_ ( aleph ` B ) ) )
9	4, 7, 8	mpsyl 68	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> om ~<_ ( aleph ` B ) )
10		fvex 6201	. . . 4 |- ( aleph ` A ) e. _V
11		ordom 7074	. . . . . 6 |- Ord om
12		2onn 7720	. . . . . 6 |- 2o e. om
13		ordelss 5739	. . . . . 6 |- ( ( Ord om /\ 2o e. om ) -> 2o C_ om )
14	11, 12, 13	mp2an 708	. . . . 5 |- 2o C_ om
15		simpll 790	. . . . . 6 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> A e. On )
16		alephgeom 8905	. . . . . 6 |- ( A e. On <-> om C_ ( aleph ` A ) )
17	15, 16	sylib 208	. . . . 5 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> om C_ ( aleph ` A ) )
18	14, 17	syl5ss 3614	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> 2o C_ ( aleph ` A ) )
19		ssdomg 8001	. . . 4 |- ( ( aleph ` A ) e. _V -> ( 2o C_ ( aleph ` A ) -> 2o ~<_ ( aleph ` A ) ) )
20	10, 18, 19	mpsyl 68	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> 2o ~<_ ( aleph ` A ) )
21		alephord3 8901	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) C_ ( aleph ` B ) ) )
22		ssdomg 8001	. . . . . . 7 |- ( ( aleph ` B ) e. _V -> ( ( aleph ` A ) C_ ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
23	4, 22	ax-mp 5	. . . . . 6 |- ( ( aleph ` A ) C_ ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
24	21, 23	syl6bi 243	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
25	24	imp 445	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
26	4	canth2 8113	. . . . 5 |- ( aleph ` B ) ~< ~P ( aleph ` B )
27		sdomdom 7983	. . . . 5 |- ( ( aleph ` B ) ~< ~P ( aleph ` B ) -> ( aleph ` B ) ~<_ ~P ( aleph ` B ) )
28	26, 27	ax-mp 5	. . . 4 |- ( aleph ` B ) ~<_ ~P ( aleph ` B )
29		domtr 8009	. . . 4 |- ( ( ( aleph ` A ) ~<_ ( aleph ` B ) /\ ( aleph ` B ) ~<_ ~P ( aleph ` B ) ) -> ( aleph ` A ) ~<_ ~P ( aleph ` B ) )
30	25, 28, 29	sylancl 694	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( aleph ` A ) ~<_ ~P ( aleph ` B ) )
31		mappwen 8935	. . 3 |- ( ( ( ( aleph ` B ) e. dom card /\ om ~<_ ( aleph ` B ) ) /\ ( 2o ~<_ ( aleph ` A ) /\ ( aleph ` A ) ~<_ ~P ( aleph ` B ) ) ) -> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ~P ( aleph ` B ) )
32	3, 9, 20, 30, 31	syl22anc 1327	. 2 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ~P ( aleph ` B ) )
33	4	pw2en 8067	. . 3 |- ~P ( aleph ` B ) ~~ ( 2o ^m ( aleph ` B ) )
34		enen2 8101	. . 3 |- ( ~P ( aleph ` B ) ~~ ( 2o ^m ( aleph ` B ) ) -> ( ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ~P ( aleph ` B ) <-> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ( 2o ^m ( aleph ` B ) ) ) )
35	33, 34	ax-mp 5	. 2 |- ( ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ~P ( aleph ` B ) <-> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ( 2o ^m ( aleph ` B ) ) )
36	32, 35	sylib 208	1 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ( 2o ^m ( aleph ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   Ord word 5722   Oncon0 5723   ` cfv 5888  (class class class)co 6650   omcom 7065   2oc2o 7554   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephexp2  9403