Theorem xpdom3 4464

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 2300	. 2 |- (B =/= (/) <-> E.x x e. B)
2		visset 1820	. . . . 5 |- x e. V
3	2	snss 2475	. . . 4 |- (x e. B <-> {x} (_ B)
4		ssid 2091	. . . . . 6 |- A (_ A
5		ssxp 3272	. . . . . 6 |- ((A (_ A /\ {x} (_ B) -> (A X. {x}) (_ (A X. B))
6	4, 5	mpan 699	. . . . 5 |- ({x} (_ B -> (A X. {x}) (_ (A X. B))
7		xpdom3.1	. . . . . . 7 |- A e. V
8		snex 2766	. . . . . . 7 |- {x} e. V
9	7, 8	xpex 3276	. . . . . 6 |- (A X. {x}) e. V
10		ssdomg 4426	. . . . . 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 4454	. . . . . . 7 |- (A X. {x}) ~~ A
13	7, 12	ensymi 4431	. . . . . 6 |- A ~~ (A X. {x})
14		endomtr 4438	. . . . . 6 |- ((A ~~ (A X. {x}) /\ (A X. {x}) ~<_ (A X. B)) -> A ~<_ (A X. B))
15	13, 14	mpan 699	. . . . 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 1299	. 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 962  E.wex 984   =/= wne 1592  Vcvv 1818   (_ wss 2058  (/)c0 2291  {csn 2421   class class class wbr 2634   X. cxp 3184   ~~ cen 4382   ~<_ cdom 4383

This theorem is referenced by:  infxpabs 7603

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-int 2548  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-f1 3211  df-fo 3212  df-f1o 3213  df-er 4277  df-en 4386  df-dom 4387

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