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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege98 | Structured version Visualization version GIF version |
Description: If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege98.x | ⊢ 𝑋 ∈ 𝐴 |
frege98.y | ⊢ 𝑌 ∈ 𝐵 |
frege98.z | ⊢ 𝑍 ∈ 𝐶 |
frege98.r | ⊢ 𝑅 ∈ 𝐷 |
Ref | Expression |
---|---|
frege98 | ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege98.x | . . . 4 ⊢ 𝑋 ∈ 𝐴 | |
2 | frege98.r | . . . 4 ⊢ 𝑅 ∈ 𝐷 | |
3 | 1, 2 | frege97 38756 | . . 3 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
4 | frege98.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege98.z | . . . 4 ⊢ 𝑍 ∈ 𝐶 | |
6 | fvex 6362 | . . . . 5 ⊢ (t+‘𝑅) ∈ V | |
7 | imaexg 7268 | . . . . 5 ⊢ ((t+‘𝑅) ∈ V → ((t+‘𝑅) “ {𝑋}) ∈ V) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ((t+‘𝑅) “ {𝑋}) ∈ V |
9 | 4, 5, 2, 8 | frege84 38743 | . . 3 ⊢ (𝑅 hereditary ((t+‘𝑅) “ {𝑋}) → (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋})))) |
10 | 3, 9 | ax-mp 5 | . 2 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋}))) |
11 | 1 | elexi 3353 | . . . 4 ⊢ 𝑋 ∈ V |
12 | 4 | elexi 3353 | . . . 4 ⊢ 𝑌 ∈ V |
13 | 11, 12 | elimasn 5648 | . . 3 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) |
14 | df-br 4805 | . . 3 ⊢ (𝑋(t+‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) | |
15 | 13, 14 | bitr4i 267 | . 2 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑌) |
16 | 5 | elexi 3353 | . . . . 5 ⊢ 𝑍 ∈ V |
17 | 11, 16 | elimasn 5648 | . . . 4 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) |
18 | df-br 4805 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑍 ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) | |
19 | 17, 18 | bitr4i 267 | . . 3 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑍) |
20 | 19 | imbi2i 325 | . 2 ⊢ ((𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
21 | 10, 15, 20 | 3imtr3i 280 | 1 ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 Vcvv 3340 {csn 4321 〈cop 4327 class class class wbr 4804 “ cima 5269 ‘cfv 6049 t+ctcl 13925 hereditary whe 38568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-frege1 38586 ax-frege2 38587 ax-frege8 38605 ax-frege52a 38653 ax-frege58b 38697 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1051 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-seq 12996 df-trcl 13927 df-relexp 13960 df-he 38569 |
This theorem is referenced by: (None) |
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