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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 43383. (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 5865 | . . 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 1376 | . 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵) |
8 | frege96d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
9 | trclfvlb 14981 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
10 | coss1 5852 | . . . . 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 14989 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
13 | 11, 12 | sstrdi 3990 | . . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
14 | 13 | ssbrd 5185 | . 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 2099 Vcvv 3470 ⊆ wss 3945 class class class wbr 5142 ∘ ccom 5676 ‘cfv 6542 t+ctcl 14958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-trcl 14960 |
This theorem is referenced by: frege87d 43174 frege102d 43178 frege129d 43187 |
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