Description: Whatever follows X in the R -sequence belongs to the R -sequence beginning with X . Proposition 106 of Frege1879 p. 73. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | frege103.z | ⊢ 𝑍 ∈ 𝑉 | |
| Assertion | frege106 | ⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege103.z | ⊢ 𝑍 ∈ 𝑉 | |
| 2 | 1 | frege105 | ⊢ ( ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) |
| 3 | frege37 | ⊢ ( ( ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) ) | |
| 4 | 2 3 | ax-mp | ⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) |