Theorem ixp0 6625

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	⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Proof of Theorem ixp0

Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		notm0 3383	. . . 4 ⊢ (¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ 𝐵 = ∅)
2	1	rexbii 2442	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnalim 2427	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
4	2, 3	sylbir 134	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
5		ixpm 6624	. . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	con3i 621	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
7		notm0 3383	. . 3 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅)
8	6, 7	sylib 121	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 𝐵 = ∅)
9	4, 8	syl 14	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ∅c0 3363  Xcixp 6592

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-nul 3364  df-ixp 6593

This theorem is referenced by: (None)