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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 43950 | . . 3 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
4 | frege98.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege98.z | . . . 4 ⊢ 𝑍 ∈ 𝐶 | |
6 | fvex 6920 | . . . . 5 ⊢ (t+‘𝑅) ∈ V | |
7 | imaexg 7936 | . . . . 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 43937 | . . 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 3501 | . . . 4 ⊢ 𝑋 ∈ V |
12 | 4 | elexi 3501 | . . . 4 ⊢ 𝑌 ∈ V |
13 | 11, 12 | elimasn 6110 | . . 3 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) |
14 | df-br 5149 | . . 3 ⊢ (𝑋(t+‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) | |
15 | 13, 14 | bitr4i 278 | . 2 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑌) |
16 | 5 | elexi 3501 | . . . . 5 ⊢ 𝑍 ∈ V |
17 | 11, 16 | elimasn 6110 | . . . 4 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) |
18 | df-br 5149 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑍 ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) | |
19 | 17, 18 | bitr4i 278 | . . 3 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑍) |
20 | 19 | imbi2i 336 | . 2 ⊢ ((𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
21 | 10, 15, 20 | 3imtr3i 291 | 1 ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 class class class wbr 5148 “ cima 5692 ‘cfv 6563 t+ctcl 15021 hereditary whe 43762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-frege1 43780 ax-frege2 43781 ax-frege8 43799 ax-frege52a 43847 ax-frege58b 43891 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-trcl 15023 df-relexp 15056 df-he 43763 |
This theorem is referenced by: (None) |
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