Theorem alephmul 9400

Description: The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
alephmul	|- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) X. ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )

Proof of Theorem alephmul

Step	Hyp	Ref	Expression
1		alephgeom 8905	. . . 4 |- ( A e. On <-> om C_ ( aleph ` A ) )
2		fvex 6201	. . . . 5 |- ( aleph ` A ) e. _V
3		ssdomg 8001	. . . . 5 |- ( ( aleph ` A ) e. _V -> ( om C_ ( aleph ` A ) -> om ~<_ ( aleph ` A ) ) )
4	2, 3	ax-mp 5	. . . 4 |- ( om C_ ( aleph ` A ) -> om ~<_ ( aleph ` A ) )
5	1, 4	sylbi 207	. . 3 |- ( A e. On -> om ~<_ ( aleph ` A ) )
6		alephon 8892	. . . 4 |- ( aleph ` A ) e. On
7		onenon 8775	. . . 4 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
8	6, 7	ax-mp 5	. . 3 |- ( aleph ` A ) e. dom card
9	5, 8	jctil 560	. 2 |- ( A e. On -> ( ( aleph ` A ) e. dom card /\ om ~<_ ( aleph ` A ) ) )
10		alephgeom 8905	. . . 4 |- ( B e. On <-> om C_ ( aleph ` B ) )
11		fvex 6201	. . . . . 6 |- ( aleph ` B ) e. _V
12		ssdomg 8001	. . . . . 6 |- ( ( aleph ` B ) e. _V -> ( om C_ ( aleph ` B ) -> om ~<_ ( aleph ` B ) ) )
13	11, 12	ax-mp 5	. . . . 5 |- ( om C_ ( aleph ` B ) -> om ~<_ ( aleph ` B ) )
14		infn0 8222	. . . . 5 |- ( om ~<_ ( aleph ` B ) -> ( aleph ` B ) =/= (/) )
15	13, 14	syl 17	. . . 4 |- ( om C_ ( aleph ` B ) -> ( aleph ` B ) =/= (/) )
16	10, 15	sylbi 207	. . 3 |- ( B e. On -> ( aleph ` B ) =/= (/) )
17		alephon 8892	. . . 4 |- ( aleph ` B ) e. On
18		onenon 8775	. . . 4 |- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card )
19	17, 18	ax-mp 5	. . 3 |- ( aleph ` B ) e. dom card
20	16, 19	jctil 560	. 2 |- ( B e. On -> ( ( aleph ` B ) e. dom card /\ ( aleph ` B ) =/= (/) ) )
21		infxp 9037	. 2 |- ( ( ( ( aleph ` A ) e. dom card /\ om ~<_ ( aleph ` A ) ) /\ ( ( aleph ` B ) e. dom card /\ ( aleph ` B ) =/= (/) ) ) -> ( ( aleph ` A ) X. ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
22	9, 20, 21	syl2an 494	1 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) X. ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cda 8990

This theorem is referenced by: (None)