Theorem xpdom3m 6679

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 6667	. . . . . . 7 |- ( ( A e. V /\ x e. B ) -> ( A X. { x } ) ~~ A )
2	1	3adant2 981	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) ~~ A )
3	2	ensymd 6629	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~~ ( A X. { x } ) )
4		xpexg 4611	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
5	4	3adant3 982	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. B ) e. _V )
6		simp3 964	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> x e. B )
7	6	snssd 3629	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } C_ B )
8		xpss2 4608	. . . . . . 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 6624	. . . . . 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 6636	. . . . 5 |- ( ( A ~~ ( A X. { x } ) /\ ( A X. { x } ) ~<_ ( A X. B ) ) -> A ~<_ ( A X. B ) )
13	3, 11, 12	syl2anc 406	. . . 4 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~<_ ( A X. B ) )
14	13	3expia 1164	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. B -> A ~<_ ( A X. B ) ) )
15	14	exlimdv 1771	. 2 |- ( ( A e. V /\ B e. W ) -> ( E. x x e. B -> A ~<_ ( A X. B ) ) )
16	15	3impia 1159	1 |- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 943   E.wex 1449   e. wcel 1461   _Vcvv 2655   C_ wss 3035   {csn 3491   class class class wbr 3893   X. cxp 4495   ~~ cen 6584   ~<_ cdom 6585

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-er 6381  df-en 6587  df-dom 6588

This theorem is referenced by: (None)