Metamath Proof Explorer
Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003)
(Revised by Mario Carneiro, 17-Nov-2014)
|
|
Ref |
Expression |
|
Hypothesis |
rankid.1 |
⊢ 𝐴 ∈ V |
|
Assertion |
rankid |
⊢ 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rankid.1 |
⊢ 𝐴 ∈ V |
2 |
|
unir1 |
⊢ ∪ ( 𝑅1 “ On ) = V |
3 |
1 2
|
eleqtrri |
⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On ) |
4 |
|
rankidb |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ) |
5 |
3 4
|
ax-mp |
⊢ 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) |