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Theorem ixp0 6528
 Description: The infinite Cartesian product of a family with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
Assertion
Ref Expression
ixp0

Proof of Theorem ixp0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 notm0 3322 . . . 4
21rexbii 2396 . . 3
3 rexnalim 2381 . . 3
42, 3sylbir 134 . 2
5 ixpm 6527 . . . 4
65con3i 600 . . 3
7 notm0 3322 . . 3
86, 7sylib 121 . 2
94, 8syl 14 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1296  wex 1433   wcel 1445  wral 2370  wrex 2371  c0 3302  cixp 6495 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-nul 3303  df-ixp 6496 This theorem is referenced by: (None)
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