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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 9893. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
ixpn0 | ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | df-ixp 8450 | . . . . 5 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
3 | 2 | abeq2i 2945 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
4 | ne0i 4297 | . . . . 5 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → 𝐵 ≠ ∅) | |
5 | 4 | ralimi 3157 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
6 | 3, 5 | simplbiim 505 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
7 | 6 | exlimiv 1922 | . 2 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
8 | 1, 7 | sylbi 218 | 1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 {cab 2796 ≠ wne 3013 ∀wral 3135 ∅c0 4288 Fn wfn 6343 ‘cfv 6348 Xcixp 8449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-dif 3936 df-nul 4289 df-ixp 8450 |
This theorem is referenced by: ixp0 8483 ac9 9893 ac9s 9903 |
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