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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 39095 | . . 3 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
4 | frege98.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege98.z | . . . 4 ⊢ 𝑍 ∈ 𝐶 | |
6 | fvex 6447 | . . . . 5 ⊢ (t+‘𝑅) ∈ V | |
7 | imaexg 7366 | . . . . 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 39082 | . . 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 3431 | . . . 4 ⊢ 𝑋 ∈ V |
12 | 4 | elexi 3431 | . . . 4 ⊢ 𝑌 ∈ V |
13 | 11, 12 | elimasn 5732 | . . 3 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) |
14 | df-br 4875 | . . 3 ⊢ (𝑋(t+‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) | |
15 | 13, 14 | bitr4i 270 | . 2 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑌) |
16 | 5 | elexi 3431 | . . . . 5 ⊢ 𝑍 ∈ V |
17 | 11, 16 | elimasn 5732 | . . . 4 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) |
18 | df-br 4875 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑍 ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) | |
19 | 17, 18 | bitr4i 270 | . . 3 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑍) |
20 | 19 | imbi2i 328 | . 2 ⊢ ((𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
21 | 10, 15, 20 | 3imtr3i 283 | 1 ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3415 {csn 4398 〈cop 4404 class class class wbr 4874 “ cima 5346 ‘cfv 6124 t+ctcl 14104 hereditary whe 38907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-frege1 38925 ax-frege2 38926 ax-frege8 38944 ax-frege52a 38992 ax-frege58b 39036 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-n0 11620 df-z 11706 df-uz 11970 df-seq 13097 df-trcl 14106 df-relexp 14139 df-he 38908 |
This theorem is referenced by: (None) |
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