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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 40318. (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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V) |
3 | frege102d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V) |
5 | frege102d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V) |
7 | frege102d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V) |
9 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶) | |
10 | frege102d.cb | . . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵) |
12 | 2, 4, 6, 8, 9, 11 | frege96d 40101 | . 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵) |
13 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V) |
14 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
15 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵) |
16 | 14, 15 | eqbrtrd 5090 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵) |
17 | 13, 16 | frege91d 40103 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵) |
18 | frege102d.ac | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
19 | 12, 17, 18 | mpjaodan 955 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 t+ctcl 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-trcl 14349 |
This theorem is referenced by: frege108d 40108 |
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