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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege122d | Structured version Visualization version GIF version |
Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 41574. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege122d.a | ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) |
frege122d.b | ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) |
Ref | Expression |
---|---|
frege122d | ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege122d.a | . . 3 ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) | |
2 | frege122d.b | . . 3 ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) | |
3 | 1, 2 | eqtr4d 2781 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
4 | 3 | olcd 871 | 1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1539 class class class wbr 5073 ‘cfv 6426 t+ctcl 14706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-cleq 2730 |
This theorem is referenced by: (None) |
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