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Metamath Proof Explorer

Theorem exeupre

Description: Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026)

Ref Expression
Assertion exeupre Could not format assertion : No typesetting found for |- ( N e. V -> ( E. m m SucMap N <-> E! m m SucMap N ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 brsucmap Could not format ( ( m e. _V /\ N e. V ) -> ( m SucMap N <-> suc m = N ) ) : No typesetting found for |- ( ( m e. _V /\ N e. V ) -> ( m SucMap N <-> suc m = N ) ) with typecode |-
2 1 el2v1 Could not format ( N e. V -> ( m SucMap N <-> suc m = N ) ) : No typesetting found for |- ( N e. V -> ( m SucMap N <-> suc m = N ) ) with typecode |-
3 2 exbidv Could not format ( N e. V -> ( E. m m SucMap N <-> E. m suc m = N ) ) : No typesetting found for |- ( N e. V -> ( E. m m SucMap N <-> E. m suc m = N ) ) with typecode |-
4 exeupre2 ⊢ ∃ m suc ⁡ m = N ↔ ∃! m suc ⁡ m = N
5 3 4 bitrdi Could not format ( N e. V -> ( E. m m SucMap N <-> E! m suc m = N ) ) : No typesetting found for |- ( N e. V -> ( E. m m SucMap N <-> E! m suc m = N ) ) with typecode |-
6 2 eubidv Could not format ( N e. V -> ( E! m m SucMap N <-> E! m suc m = N ) ) : No typesetting found for |- ( N e. V -> ( E! m m SucMap N <-> E! m suc m = N ) ) with typecode |-
7 5 6 bitr4d Could not format ( N e. V -> ( E. m m SucMap N <-> E! m m SucMap N ) ) : No typesetting found for |- ( N e. V -> ( E. m m SucMap N <-> E! m m SucMap N ) ) with typecode |-