Theorem xpdom3m 7085

Description: A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)

Assertion

Ref	Expression
xpdom3m	|- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )

Distinct variable groups:   x, A   x, B   x, V   x, W

Proof of Theorem xpdom3m

Step	Hyp	Ref	Expression
1		xpsneng 7073	. . . . . . 7 |- ( ( A e. V /\ x e. B ) -> ( A X. { x } ) ~~ A )
2	1	3adant2 1043	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) ~~ A )
3	2	ensymd 7023	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~~ ( A X. { x } ) )
4		xpexg 4864	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
5	4	3adant3 1044	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. B ) e. _V )
6		simp3 1026	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> x e. B )
7	6	snssd 3839	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } C_ B )
8		xpss2 4861	. . . . . . 7 |- ( { x } C_ B -> ( A X. { x } ) C_ ( A X. B ) )
9	7, 8	syl 14	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) C_ ( A X. B ) )
10		ssdomg 7018	. . . . . 6 |- ( ( A X. B ) e. _V -> ( ( A X. { x } ) C_ ( A X. B ) -> ( A X. { x } ) ~<_ ( A X. B ) ) )
11	5, 9, 10	sylc 62	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) ~<_ ( A X. B ) )
12		endomtr 7030	. . . . 5 |- ( ( A ~~ ( A X. { x } ) /\ ( A X. { x } ) ~<_ ( A X. B ) ) -> A ~<_ ( A X. B ) )
13	3, 11, 12	syl2anc 411	. . . 4 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~<_ ( A X. B ) )
14	13	3expia 1232	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. B -> A ~<_ ( A X. B ) ) )
15	14	exlimdv 1868	. 2 |- ( ( A e. V /\ B e. W ) -> ( E. x x e. B -> A ~<_ ( A X. B ) ) )
16	15	3impia 1227	1 |- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   E.wex 1541   e. wcel 2203   _Vcvv 2813   C_ wss 3211   {csn 3689   class class class wbr 4109   X. cxp 4747   ~~ cen 6973   ~<_ cdom 6974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-er 6767  df-en 6976  df-dom 6977

This theorem is referenced by: (None)