Theorem ixp0x 4343

Description: An infinite Cartesian product with an empty index set.

Assertion

Ref	Expression
ixp0x	|- X_x e. (/) A = {(/)}

Proof of Theorem ixp0x

Step	Hyp	Ref	Expression
1		df-ixp 4332	. 2 |- X_x e. (/) A = {f | (f Fn (/) /\ A.x e. (/) (f` x) e. A)}
2		elsn 2411	. . . 4 |- (f e. {(/)} <-> f = (/))
3		fn0 3591	. . . 4 |- (f Fn (/) <-> f = (/))
4		ral0 2348	. . . . 5 |- A.x e. (/) (f` x) e. A
5	4	biantru 722	. . . 4 |- (f Fn (/) <-> (f Fn (/) /\ A.x e. (/) (f` x) e. A))
6	2, 3, 5	3bitr2 179	. . 3 |- (f e. {(/)} <-> (f Fn (/) /\ A.x e. (/) (f` x) e. A))
7	6	abbi2i 1566	. 2 |- {(/)} = {f | (f Fn (/) /\ A.x e. (/) (f` x) e. A)}
8	1, 7	eqtr4 1490	1 |- X_x e. (/) A = {(/)}

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637  (/)c0 2270  {csn 2399   Fn wfn 3167  ` cfv 3172  X_cixp 4331

This theorem is referenced by:  0elixp 4344

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-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-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-fun 3182  df-fn 3183  df-ixp 4332

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