Theorem ixpn0 8924

Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 10478. (Contributed by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
ixpn0	⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Proof of Theorem ixpn0

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4347	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
2		df-ixp 8892	. . . . 5 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
3	2	eqabri 2878	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
4		ne0i 4335	. . . . 5 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → 𝐵 ≠ ∅)
5	4	ralimi 3084	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
6	3, 5	simplbiim 506	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
7	6	exlimiv 1934	. 2 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
8	1, 7	sylbi 216	1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 397  ∃wex 1782   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∅c0 4323   Fn wfn 6539  ‘cfv 6544  Xcixp 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-dif 3952  df-nul 4324  df-ixp 8892

This theorem is referenced by:  ixp0  8925  ac9  10478  ac9s  10488