Metamath Proof Explorer


Theorem grusucd

Description: Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023)

Ref Expression
Hypotheses grusucd.1 ( 𝜑𝐺 ∈ Univ )
grusucd.2 ( 𝜑𝐴𝐺 )
Assertion grusucd ( 𝜑 → suc 𝐴𝐺 )

Proof

Step Hyp Ref Expression
1 grusucd.1 ( 𝜑𝐺 ∈ Univ )
2 grusucd.2 ( 𝜑𝐴𝐺 )
3 df-suc suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
4 grusn ( ( 𝐺 ∈ Univ ∧ 𝐴𝐺 ) → { 𝐴 } ∈ 𝐺 )
5 1 2 4 syl2anc ( 𝜑 → { 𝐴 } ∈ 𝐺 )
6 gruun ( ( 𝐺 ∈ Univ ∧ 𝐴𝐺 ∧ { 𝐴 } ∈ 𝐺 ) → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝐺 )
7 1 2 5 6 syl3anc ( 𝜑 → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝐺 )
8 3 7 eqeltrid ( 𝜑 → suc 𝐴𝐺 )