Metamath Proof Explorer
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypothesis |
bnj216.1 |
⊢ 𝐵 ∈ V |
|
Assertion |
bnj216 |
⊢ ( 𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bnj216.1 |
⊢ 𝐵 ∈ V |
2 |
1
|
sucid |
⊢ 𝐵 ∈ suc 𝐵 |
3 |
|
eleq2 |
⊢ ( 𝐴 = suc 𝐵 → ( 𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵 ) ) |
4 |
2 3
|
mpbiri |
⊢ ( 𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴 ) |