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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 38259. (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 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V) |
3 | frege102d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V) |
5 | frege102d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V) |
7 | frege102d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) | |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V) |
9 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶) | |
10 | frege102d.cb | . . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵) |
12 | 2, 4, 6, 8, 9, 11 | frege96d 38041 | . 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵) |
13 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V) |
14 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
15 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵) |
16 | 14, 15 | eqbrtrd 4675 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵) |
17 | 13, 16 | frege91d 38043 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵) |
18 | frege102d.ac | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
19 | 12, 17, 18 | mpjaodan 827 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 t+ctcl 13724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-trcl 13726 |
This theorem is referenced by: frege108d 38048 |
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