Description: Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | grusucd.1 | ⊢ ( 𝜑 → 𝐺 ∈ Univ ) | |
grusucd.2 | ⊢ ( 𝜑 → 𝐴 ∈ 𝐺 ) | ||
Assertion | grusucd | ⊢ ( 𝜑 → suc 𝐴 ∈ 𝐺 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grusucd.1 | ⊢ ( 𝜑 → 𝐺 ∈ Univ ) | |
2 | grusucd.2 | ⊢ ( 𝜑 → 𝐴 ∈ 𝐺 ) | |
3 | df-suc | ⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) | |
4 | grusn | ⊢ ( ( 𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ) → { 𝐴 } ∈ 𝐺 ) | |
5 | 1 2 4 | syl2anc | ⊢ ( 𝜑 → { 𝐴 } ∈ 𝐺 ) |
6 | gruun | ⊢ ( ( 𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ { 𝐴 } ∈ 𝐺 ) → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝐺 ) | |
7 | 1 2 5 6 | syl3anc | ⊢ ( 𝜑 → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝐺 ) |
8 | 3 7 | eqeltrid | ⊢ ( 𝜑 → suc 𝐴 ∈ 𝐺 ) |