Description: Shorter proof of prfi using ax-un . (Contributed by NM, 22-Aug-2008) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | prfiALT | ⊢ { 𝐴 , 𝐵 } ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr | ⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } ) | |
2 | snfi | ⊢ { 𝐴 } ∈ Fin | |
3 | snfi | ⊢ { 𝐵 } ∈ Fin | |
4 | unfi | ⊢ ( ( { 𝐴 } ∈ Fin ∧ { 𝐵 } ∈ Fin ) → ( { 𝐴 } ∪ { 𝐵 } ) ∈ Fin ) | |
5 | 2 3 4 | mp2an | ⊢ ( { 𝐴 } ∪ { 𝐵 } ) ∈ Fin |
6 | 1 5 | eqeltri | ⊢ { 𝐴 , 𝐵 } ∈ Fin |