Description: The class of elements of A "such that A is a set" is a set. That class is equal to A when A is a set (see class2seteq ) and to the empty set when A is a proper class. (Contributed by NM, 16-Oct-2003)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | class2set | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabexg | ⊢ ( 𝐴 ∈ V → { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } ∈ V ) | |
| 2 | simpl | ⊢ ( ( ¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝐴 ∈ V ) | |
| 3 | 2 | nrexdv | ⊢ ( ¬ 𝐴 ∈ V → ¬ ∃ 𝑥 ∈ 𝐴 𝐴 ∈ V ) |
| 4 | rabn0 | ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝐴 ∈ V ) | |
| 5 | 4 | necon1bbii | ⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 𝐴 ∈ V ↔ { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } = ∅ ) |
| 6 | 3 5 | sylib | ⊢ ( ¬ 𝐴 ∈ V → { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } = ∅ ) |
| 7 | 0ex | ⊢ ∅ ∈ V | |
| 8 | 6 7 | eqeltrdi | ⊢ ( ¬ 𝐴 ∈ V → { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } ∈ V ) |
| 9 | 1 8 | pm2.61i | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V } ∈ V |