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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 43421 | . . 3 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
4 | frege98.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege98.z | . . . 4 ⊢ 𝑍 ∈ 𝐶 | |
6 | fvex 6915 | . . . . 5 ⊢ (t+‘𝑅) ∈ V | |
7 | imaexg 7927 | . . . . 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 43408 | . . 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 3493 | . . . 4 ⊢ 𝑋 ∈ V |
12 | 4 | elexi 3493 | . . . 4 ⊢ 𝑌 ∈ V |
13 | 11, 12 | elimasn 6098 | . . 3 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ (t+‘𝑅)) |
14 | df-br 5153 | . . 3 ⊢ (𝑋(t+‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (t+‘𝑅)) | |
15 | 13, 14 | bitr4i 277 | . 2 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑌) |
16 | 5 | elexi 3493 | . . . . 5 ⊢ 𝑍 ∈ V |
17 | 11, 16 | elimasn 6098 | . . . 4 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ ⟨𝑋, 𝑍⟩ ∈ (t+‘𝑅)) |
18 | df-br 5153 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑍 ↔ ⟨𝑋, 𝑍⟩ ∈ (t+‘𝑅)) | |
19 | 17, 18 | bitr4i 277 | . . 3 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 𝑋(t+‘𝑅)𝑍) |
20 | 19 | imbi2i 335 | . 2 ⊢ ((𝑌(t+‘𝑅)𝑍 → 𝑍 ∈ ((t+‘𝑅) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
21 | 10, 15, 20 | 3imtr3i 290 | 1 ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3473 {csn 4632 ⟨cop 4638 class class class wbr 5152 “ cima 5685 ‘cfv 6553 t+ctcl 14972 hereditary whe 43233 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-frege1 43251 ax-frege2 43252 ax-frege8 43270 ax-frege52a 43318 ax-frege58b 43362 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14007 df-trcl 14974 df-relexp 15007 df-he 43234 |
This theorem is referenced by: (None) |
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