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Theorem ixp0 6634
 Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
Assertion
Ref Expression
ixp0 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)

Proof of Theorem ixp0
Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 notm0 3389 . . . 4 (¬ ∃𝑧 𝑧𝐵𝐵 = ∅)
21rexbii 2446 . . 3 (∃𝑥𝐴 ¬ ∃𝑧 𝑧𝐵 ↔ ∃𝑥𝐴 𝐵 = ∅)
3 rexnalim 2428 . . 3 (∃𝑥𝐴 ¬ ∃𝑧 𝑧𝐵 → ¬ ∀𝑥𝐴𝑧 𝑧𝐵)
42, 3sylbir 134 . 2 (∃𝑥𝐴 𝐵 = ∅ → ¬ ∀𝑥𝐴𝑧 𝑧𝐵)
5 ixpm 6633 . . . 4 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
65con3i 622 . . 3 (¬ ∀𝑥𝐴𝑧 𝑧𝐵 → ¬ ∃𝑓 𝑓X𝑥𝐴 𝐵)
7 notm0 3389 . . 3 (¬ ∃𝑓 𝑓X𝑥𝐴 𝐵X𝑥𝐴 𝐵 = ∅)
86, 7sylib 121 . 2 (¬ ∀𝑥𝐴𝑧 𝑧𝐵X𝑥𝐴 𝐵 = ∅)
94, 8syl 14 1 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∅c0 3369  Xcixp 6601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-dif 3079  df-nul 3370  df-ixp 6602 This theorem is referenced by: (None)
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