Theorem xpdom3 5504

Description: A set is dominated by its cross product with a non-empty set. Exercise 6 of [Suppes] p. 98.

Hypothesis

Ref	Expression
xpdom3.1	|- A e. _V

Assertion

Ref	Expression
xpdom3	|- (B =/= (/) -> A ~<_ (A X. B))

Proof of Theorem xpdom3

Step	Hyp	Ref	Expression
1		n0 2884	. 2 |- (B =/= (/) <-> E.x x e. B)
2		visset 2295	. . . . 5 |- x e. _V
3	2	snss 3122	. . . 4 |- (x e. B <-> {x} C_ B)
4		ssid 2634	. . . . . 6 |- A C_ A
5		xpss12 4089	. . . . . 6 |- ((A C_ A /\ {x} C_ B) -> (A X. {x}) C_ (A X. B))
6	4, 5	mpan 759	. . . . 5 |- ({x} C_ B -> (A X. {x}) C_ (A X. B))
7		xpdom3.1	. . . . . . 7 |- A e. _V
8		snex 3492	. . . . . . 7 |- {x} e. _V
9	7, 8	xpex 4096	. . . . . 6 |- (A X. {x}) e. _V
10		ssdomg 5467	. . . . . 6 |- ((A X. {x}) e. _V -> ((A X. {x}) C_ (A X. B) -> (A X. {x}) ~<_ (A X. B)))
11	9, 10	ax-mp 7	. . . . 5 |- ((A X. {x}) C_ (A X. B) -> (A X. {x}) ~<_ (A X. B))
12	7, 2	xpsnen 5494	. . . . . . 7 |- (A X. {x}) ~~ A
13	7, 12	ensymi 5472	. . . . . 6 |- A ~~ (A X. {x})
14		endomtr 5479	. . . . . 6 |- ((A ~~ (A X. {x}) /\ (A X. {x}) ~<_ (A X. B)) -> A ~<_ (A X. B))
15	13, 14	mpan 759	. . . . 5 |- ((A X. {x}) ~<_ (A X. B) -> A ~<_ (A X. B))
16	6, 11, 15	3syl 24	. . . 4 |- ({x} C_ B -> A ~<_ (A X. B))
17	3, 16	sylbi 216	. . 3 |- (x e. B -> A ~<_ (A X. B))
18	17	19.23aiv 1674	. 2 |- (E.x x e. B -> A ~<_ (A X. B))
19	1, 18	sylbi 216	1 |- (B =/= (/) -> A ~<_ (A X. B))

Syntax hints:   -> wi 3   e. wcel 1300  E.wex 1326   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338   X. cxp 3984   ~~ cen 5423   ~<_ cdom 5424

This theorem is referenced by:  xpfi 5632  infxpabs 8839

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-dom 5428

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