Metamath Proof Explorer


Theorem wunsuc

Description: A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wununi.1 ( 𝜑𝑈 ∈ WUni )
wununi.2 ( 𝜑𝐴𝑈 )
Assertion wunsuc ( 𝜑 → suc 𝐴𝑈 )

Proof

Step Hyp Ref Expression
1 wununi.1 ( 𝜑𝑈 ∈ WUni )
2 wununi.2 ( 𝜑𝐴𝑈 )
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 𝐴𝑈 )