Theorem xpdom3m 6776

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 6764	. . . . . . 7 |- ( ( A e. V /\ x e. B ) -> ( A X. { x } ) ~~ A )
2	1	3adant2 1001	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. { x } ) ~~ A )
3	2	ensymd 6725	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~~ ( A X. { x } ) )
4		xpexg 4699	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
5	4	3adant3 1002	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A X. B ) e. _V )
6		simp3 984	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> x e. B )
7	6	snssd 3701	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } C_ B )
8		xpss2 4696	. . . . . . 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 6720	. . . . . 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 6732	. . . . 5 |- ( ( A ~~ ( A X. { x } ) /\ ( A X. { x } ) ~<_ ( A X. B ) ) -> A ~<_ ( A X. B ) )
13	3, 11, 12	syl2anc 409	. . . 4 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~<_ ( A X. B ) )
14	13	3expia 1187	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. B -> A ~<_ ( A X. B ) ) )
15	14	exlimdv 1799	. 2 |- ( ( A e. V /\ B e. W ) -> ( E. x x e. B -> A ~<_ ( A X. B ) ) )
16	15	3impia 1182	1 |- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   E.wex 1472   e. wcel 2128   _Vcvv 2712   C_ wss 3102   {csn 3560   class class class wbr 3965   X. cxp 4583   ~~ cen 6680   ~<_ cdom 6681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-er 6477  df-en 6683  df-dom 6684

This theorem is referenced by: (None)