![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 39092. (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 5527 | . . 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 1501 | . 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵) |
8 | frege96d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
9 | trclfvlb 14133 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
10 | coss1 5514 | . . . . 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 14141 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
13 | 11, 12 | syl6ss 3839 | . . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
14 | 13 | ssbrd 4918 | . 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 2164 Vcvv 3414 ⊆ wss 3798 class class class wbr 4875 ∘ ccom 5350 ‘cfv 6127 t+ctcl 14110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-iota 6090 df-fun 6129 df-fv 6135 df-trcl 14112 |
This theorem is referenced by: frege87d 38882 frege102d 38886 frege129d 38895 |
Copyright terms: Public domain | W3C validator |