Description: The internal direct product of a family of subgroups indexed by a proper class is empty. (Contributed by AV, 13-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | dprdval0prc | ⊢ ( dom 𝑆 ∉ V → ( 𝐺 DProd 𝑆 ) = ∅ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel | ⊢ ( dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V ) | |
2 | dmexg | ⊢ ( 𝑆 ∈ V → dom 𝑆 ∈ V ) | |
3 | 2 | con3i | ⊢ ( ¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V ) |
4 | 1 3 | sylbi | ⊢ ( dom 𝑆 ∉ V → ¬ 𝑆 ∈ V ) |
5 | reldmdprd | ⊢ Rel dom DProd | |
6 | 5 | ovprc2 | ⊢ ( ¬ 𝑆 ∈ V → ( 𝐺 DProd 𝑆 ) = ∅ ) |
7 | 4 6 | syl | ⊢ ( dom 𝑆 ∉ V → ( 𝐺 DProd 𝑆 ) = ∅ ) |