Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege108d | Structured version Visualization version GIF version |
Description: If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 40310. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege108d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege108d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege108d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege108d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
frege108d.ac | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) |
frege108d.cb | ⊢ (𝜑 → 𝐶𝑅𝐵) |
Ref | Expression |
---|---|
frege108d | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege108d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | frege108d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
3 | frege108d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
4 | frege108d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
5 | frege108d.ac | . . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
6 | frege108d.cb | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | frege102d 40092 | . 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
8 | 7 | frege106d 40093 | 1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ‘cfv 6349 t+ctcl 14339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-trcl 14341 |
This theorem is referenced by: frege111d 40097 |
Copyright terms: Public domain | W3C validator |