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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 43587. (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 43370 | . 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
8 | 7 | frege106d 43371 | 1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 Vcvv 3461 class class class wbr 5152 ‘cfv 6553 t+ctcl 14985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-iota 6505 df-fun 6555 df-fv 6561 df-trcl 14987 |
This theorem is referenced by: frege111d 43375 |
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