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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege96d | Structured version Visualization version GIF version |
Description: If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 40660. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege96d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege96d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege96d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege96d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
frege96d.ac | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) |
frege96d.cb | ⊢ (𝜑 → 𝐶𝑅𝐵) |
Ref | Expression |
---|---|
frege96d | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege96d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | frege96d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | frege96d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
4 | frege96d.ac | . . 3 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) | |
5 | frege96d.cb | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
6 | brcogw 5703 | . . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶 ∧ 𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | syl32anc 1375 | . 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵) |
8 | frege96d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
9 | trclfvlb 14359 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
10 | coss1 5690 | . . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅))) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅))) |
12 | trclfvcotrg 14367 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
13 | 11, 12 | sstrdi 3927 | . . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
14 | 13 | ssbrd 5073 | . 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ (t+‘𝑅))𝐵 → 𝐴(t+‘𝑅)𝐵)) |
15 | 7, 14 | mpd 15 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ∘ ccom 5523 ‘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: frege87d 40451 frege102d 40455 frege129d 40464 |
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