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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 10521. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
ixpn0 | ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4359 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
2 | df-ixp 8937 | . . . . 5 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
3 | 2 | eqabri 2883 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
4 | ne0i 4347 | . . . . 5 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → 𝐵 ≠ ∅) | |
5 | 4 | ralimi 3081 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
6 | 3, 5 | simplbiim 504 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
7 | 6 | exlimiv 1928 | . 2 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
8 | 1, 7 | sylbi 217 | 1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∀wral 3059 ∅c0 4339 Fn wfn 6558 ‘cfv 6563 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-dif 3966 df-nul 4340 df-ixp 8937 |
This theorem is referenced by: ixp0 8970 ac9 10521 ac9s 10531 |
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