Theorem xpdom3m 7122

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 7110	. . . . . . 7 |- ( ( A e. V /\ x e. B ) -> ( A X. { x } ) ~~ A )
2	1	3adant2 1047	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) ~~ A )
3	2	ensymd 7060	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~~ ( A X. { x } ) )
4		xpexg 4884	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
5	4	3adant3 1048	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. B ) e. _V )
6		simp3 1030	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> x e. B )
7	6	snssd 3855	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } C_ B )
8		xpss2 4881	. . . . . . 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 7055	. . . . . 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 7067	. . . . 5 |- ( ( A ~~ ( A X. { x } ) /\ ( A X. { x } ) ~<_ ( A X. B ) ) -> A ~<_ ( A X. B ) )
13	3, 11, 12	syl2anc 415	. . . 4 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~<_ ( A X. B ) )
14	13	3expia 1236	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. B -> A ~<_ ( A X. B ) ) )
15	14	exlimdv 1872	. 2 |- ( ( A e. V /\ B e. W ) -> ( E. x x e. B -> A ~<_ ( A X. B ) ) )
16	15	3impia 1231	1 |- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {csn 3705   class class class wbr 4125   X. cxp 4767   ~~ cen 7010   ~<_ cdom 7011

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-er 6797  df-en 7013  df-dom 7014

This theorem is referenced by: (None)