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Mirrors > Home > MPE Home > Th. List > ixpn0 | Structured version Visualization version GIF version |
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 10285. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
ixpn0 | ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4286 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | df-ixp 8717 | . . . . 5 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
3 | 2 | abeq2i 2873 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
4 | ne0i 4274 | . . . . 5 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → 𝐵 ≠ ∅) | |
5 | 4 | ralimi 3083 | . . . 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 1931 | . 2 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1779 ∈ wcel 2104 {cab 2713 ≠ wne 2941 ∀wral 3062 ∅c0 4262 Fn wfn 6453 ‘cfv 6458 Xcixp 8716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-dif 3895 df-nul 4263 df-ixp 8717 |
This theorem is referenced by: ixp0 8750 ac9 10285 ac9s 10295 |
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