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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege102d | 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 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 43978. (Contributed by RP, 15-Jul-2020.) |
| Ref | Expression |
|---|---|
| frege102d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
| frege102d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
| frege102d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
| frege102d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
| frege102d.ac | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) |
| frege102d.cb | ⊢ (𝜑 → 𝐶𝑅𝐵) |
| Ref | Expression |
|---|---|
| frege102d | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege102d.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ V) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V) |
| 3 | frege102d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V) |
| 5 | frege102d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
| 6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V) |
| 7 | frege102d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) | |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V) |
| 9 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶) | |
| 10 | frege102d.cb | . . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵) |
| 12 | 2, 4, 6, 8, 9, 11 | frege96d 43762 | . 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵) |
| 13 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V) |
| 14 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
| 15 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵) |
| 16 | 14, 15 | eqbrtrd 5165 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵) |
| 17 | 13, 16 | frege91d 43764 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵) |
| 18 | frege102d.ac | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
| 19 | 12, 17, 18 | mpjaodan 961 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 t+ctcl 15024 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-trcl 15026 |
| This theorem is referenced by: frege108d 43769 |
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