Theorem xpdom3 4425

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		ne0 2278	. 2 |- (B =/= (/) <-> E.x x e. B)
2		visset 1804	. . . . 5 |- x e. V
3	2	snss 2452	. . . 4 |- (x e. B <-> {x} (_ B)
4		ssid 2070	. . . . . 6 |- A (_ A
5		ssxp 3246	. . . . . 6 |- ((A (_ A /\ {x} (_ B) -> (A X. {x}) (_ (A X. B))
6	4, 5	mpan 693	. . . . 5 |- ({x} (_ B -> (A X. {x}) (_ (A X. B))
7		xpdom3.1	. . . . . . 7 |- A e. V
8		snex 2740	. . . . . . 7 |- {x} e. V
9	7, 8	xpex 3250	. . . . . 6 |- (A X. {x}) e. V
10		ssdomg 4389	. . . . . 6 |- ((A X. {x}) e. V -> ((A X. {x}) (_ (A X. B) -> (A X. {x}) ~<_ (A X. B)))
11	9, 10	ax-mp 7	. . . . 5 |- ((A X. {x}) (_ (A X. B) -> (A X. {x}) ~<_ (A X. B))
12	7, 2	xpsnen 4415	. . . . . . 7 |- (A X. {x}) ~~ A
13	7, 12	ensymi 4394	. . . . . 6 |- A ~~ (A X. {x})
14		endomtr 4401	. . . . . 6 |- ((A ~~ (A X. {x}) /\ (A X. {x}) ~<_ (A X. B)) -> A ~<_ (A X. B))
15	13, 14	mpan 693	. . . . 5 |- ((A X. {x}) ~<_ (A X. B) -> A ~<_ (A X. B))
16	6, 11, 15	3syl 20	. . . 4 |- ({x} (_ B -> A ~<_ (A X. B))
17	3, 16	sylbi 199	. . 3 |- (x e. B -> A ~<_ (A X. B))
18	17	19.23aiv 1290	. 2 |- (E.x x e. B -> A ~<_ (A X. B))
19	1, 18	sylbi 199	1 |- (B =/= (/) -> A ~<_ (A X. B))

Syntax hints:   -> wi 3   e. wcel 955  E.wex 977   =/= wne 1577  Vcvv 1802   (_ wss 2037  (/)c0 2270  {csn 2399   class class class wbr 2609   X. cxp 3158   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  infxpabs 7513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-er 4245  df-en 4351  df-dom 4352

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