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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 37273. (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 5212 | . . 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 1326 | . 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵) |
8 | frege96d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
9 | trclfvlb 13597 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
10 | coss1 5199 | . . . . 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 13605 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
13 | 11, 12 | syl6ss 3580 | . . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
14 | 13 | ssbrd 4626 | . 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 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ∘ ccom 5042 ‘cfv 5804 t+ctcl 13572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-trcl 13574 |
This theorem is referenced by: frege87d 37061 frege102d 37065 frege129d 37074 |
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