Metamath Proof Explorer
Description: A weak universe is closed under successors. (Contributed by Mario
Carneiro, 2-Jan-2017)
|
|
Ref |
Expression |
|
Hypotheses |
wununi.1 |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
|
|
wununi.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
|
Assertion |
wunsuc |
⊢ ( 𝜑 → suc 𝐴 ∈ 𝑈 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
wununi.1 |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
2 |
|
wununi.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
3 |
|
df-suc |
⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) |
4 |
1 2
|
wunsn |
⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 ) |
5 |
1 2 4
|
wunun |
⊢ ( 𝜑 → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝑈 ) |
6 |
3 5
|
eqeltrid |
⊢ ( 𝜑 → suc 𝐴 ∈ 𝑈 ) |