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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege111d | 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 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 43965. (Contributed by RP, 15-Jul-2020.) |
| Ref | Expression |
|---|---|
| frege111d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
| frege111d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
| frege111d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
| frege111d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
| frege111d.ac | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) |
| frege111d.cb | ⊢ (𝜑 → 𝐶𝑅𝐵) |
| Ref | Expression |
|---|---|
| frege111d | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege111d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
| 2 | frege111d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
| 3 | frege111d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
| 4 | frege111d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
| 5 | frege111d.ac | . . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
| 6 | frege111d.cb | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
| 7 | 1, 2, 3, 4, 5, 6 | frege108d 43747 | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) |
| 8 | 7 | frege114d 43749 | 1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 Vcvv 3479 class class class wbr 5141 ‘cfv 6559 t+ctcl 15020 |
| 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 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-iota 6512 df-fun 6561 df-fv 6567 df-trcl 15022 |
| This theorem is referenced by: (None) |
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