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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 43973 | . . 3 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
| 4 | frege98.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
| 5 | frege98.z | . . . 4 ⊢ 𝑍 ∈ 𝐶 | |
| 6 | fvex 6919 | . . . . 5 ⊢ (t+‘𝑅) ∈ V | |
| 7 | imaexg 7935 | . . . . 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 43960 | . . 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 3503 | . . . 4 ⊢ 𝑋 ∈ V |
| 12 | 4 | elexi 3503 | . . . 4 ⊢ 𝑌 ∈ V |
| 13 | 11, 12 | elimasn 6108 | . . 3 ⊢ (𝑌 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ (t+‘𝑅)) |
| 14 | df-br 5144 | . . 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 3503 | . . . . 5 ⊢ 𝑍 ∈ V |
| 17 | 11, 16 | elimasn 6108 | . . . 4 ⊢ (𝑍 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑍〉 ∈ (t+‘𝑅)) |
| 18 | df-br 5144 | . . . 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 2108 Vcvv 3480 {csn 4626 〈cop 4632 class class class wbr 5143 “ cima 5688 ‘cfv 6561 t+ctcl 15024 hereditary whe 43785 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-frege1 43803 ax-frege2 43804 ax-frege8 43822 ax-frege52a 43870 ax-frege58b 43914 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-trcl 15026 df-relexp 15059 df-he 43786 |
| This theorem is referenced by: (None) |
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