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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 9188. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
ixpn0 | ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3890 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | df-ixp 7795 | . . . . . 6 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
3 | 2 | abeq2i 2722 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
4 | 3 | simprbi 479 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) |
5 | ne0i 3880 | . . . . 5 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → 𝐵 ≠ ∅) | |
6 | 5 | ralimi 2936 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
8 | 7 | exlimiv 1845 | . 2 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
9 | 1, 8 | sylbi 206 | 1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∅c0 3874 Fn wfn 5799 ‘cfv 5804 Xcixp 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-nul 3875 df-ixp 7795 |
This theorem is referenced by: ixp0 7827 ac9 9188 ac9s 9198 |
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