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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 40666. (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 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V) |
3 | frege102d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
4 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V) |
5 | frege102d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V) |
7 | frege102d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) | |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V) |
9 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶) | |
10 | frege102d.cb | . . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵) |
12 | 2, 4, 6, 8, 9, 11 | frege96d 40450 | . 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵) |
13 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V) |
14 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
15 | 10 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵) |
16 | 14, 15 | eqbrtrd 5052 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵) |
17 | 13, 16 | frege91d 40452 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵) |
18 | frege102d.ac | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
19 | 12, 17, 18 | mpjaodan 956 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 t+ctcl 14336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-trcl 14338 |
This theorem is referenced by: frege108d 40457 |
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