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 . (Contributed by Mario Carneiro, 22-Jun-2016)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ixpn0 | ⊢ ( X 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | ⊢ ( X 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ) | |
| 2 | df-ixp | ⊢ X 𝑥 ∈ 𝐴 𝐵 = { 𝑓 ∣ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) } | |
| 3 | 2 | abeq2i | ⊢ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
| 4 | ne0i | ⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 → 𝐵 ≠ ∅ ) | |
| 5 | 4 | ralimi | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 → ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |
| 6 | 3 5 | simplbiim | ⊢ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 → ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |
| 7 | 6 | exlimiv | ⊢ ( ∃ 𝑓 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 → ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |
| 8 | 1 7 | sylbi | ⊢ ( X 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ) |