Description: A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | omsucne | ⊢ ( 𝐴 ∈ ω → 𝐴 ≠ suc 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord | ⊢ ( 𝐴 ∈ ω → Ord 𝐴 ) | |
2 | orddisj | ⊢ ( Ord 𝐴 → ( 𝐴 ∩ { 𝐴 } ) = ∅ ) | |
3 | 1 2 | syl | ⊢ ( 𝐴 ∈ ω → ( 𝐴 ∩ { 𝐴 } ) = ∅ ) |
4 | snnzg | ⊢ ( 𝐴 ∈ ω → { 𝐴 } ≠ ∅ ) | |
5 | disjpss | ⊢ ( ( ( 𝐴 ∩ { 𝐴 } ) = ∅ ∧ { 𝐴 } ≠ ∅ ) → 𝐴 ⊊ ( 𝐴 ∪ { 𝐴 } ) ) | |
6 | 3 4 5 | syl2anc | ⊢ ( 𝐴 ∈ ω → 𝐴 ⊊ ( 𝐴 ∪ { 𝐴 } ) ) |
7 | 6 | pssned | ⊢ ( 𝐴 ∈ ω → 𝐴 ≠ ( 𝐴 ∪ { 𝐴 } ) ) |
8 | df-suc | ⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) | |
9 | 8 | neeq2i | ⊢ ( 𝐴 ≠ suc 𝐴 ↔ 𝐴 ≠ ( 𝐴 ∪ { 𝐴 } ) ) |
10 | 7 9 | sylibr | ⊢ ( 𝐴 ∈ ω → 𝐴 ≠ suc 𝐴 ) |