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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege98d | Structured version Visualization version GIF version |
Description: If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 43394. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege98d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege98d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege98d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
frege98d.ac | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) |
frege98d.cb | ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵) |
Ref | Expression |
---|---|
frege98d | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege98d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | frege98d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | frege98d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
4 | frege98d.ac | . . 3 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) | |
5 | frege98d.cb | . . 3 ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵) | |
6 | brcogw 5873 | . . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶 ∧ 𝐶(t+‘𝑅)𝐵)) → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | syl32anc 1375 | . 2 ⊢ (𝜑 → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵) |
8 | trclfvcotrg 15001 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
10 | 9 | ssbrd 5193 | . 2 ⊢ (𝜑 → (𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵 → 𝐴(t+‘𝑅)𝐵)) |
11 | 7, 10 | mpd 15 | 1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 class class class wbr 5150 ∘ ccom 5684 ‘cfv 6551 t+ctcl 14970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-iota 6503 df-fun 6553 df-fv 6559 df-trcl 14972 |
This theorem is referenced by: (None) |
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