Description: Generalization of the Axiom of Choice to proper classes. B is a collection B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ac6s4.1 | ⊢ 𝐴 ∈ V | |
| Assertion | ac6s4 | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ac6s4.1 | ⊢ 𝐴 ∈ V | |
| 2 | n0 | ⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝐵 ) | |
| 3 | 2 | ralbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝐵 ) |
| 4 | eleq1 | ⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝑦 ∈ 𝐵 ↔ ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) | |
| 5 | 1 4 | ac6s2 | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝐵 → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
| 6 | 3 5 | sylbi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |