![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege97 | Structured version Visualization version GIF version |
Description: The property of following
𝑋
in the 𝑅-sequence is hereditary
in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71.
Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege97.x | ⊢ 𝑋 ∈ 𝑈 |
frege97.r | ⊢ 𝑅 ∈ 𝑊 |
Ref | Expression |
---|---|
frege97 | ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege75 38734 | . 2 ⊢ (∀𝑏(𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) → 𝑅 hereditary ((t+‘𝑅) “ {𝑋})) | |
2 | frege97.x | . . . . 5 ⊢ 𝑋 ∈ 𝑈 | |
3 | vex 3343 | . . . . 5 ⊢ 𝑏 ∈ V | |
4 | vex 3343 | . . . . 5 ⊢ 𝑎 ∈ V | |
5 | frege97.r | . . . . 5 ⊢ 𝑅 ∈ 𝑊 | |
6 | 2, 3, 4, 5 | frege96 38755 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 → (𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎)) |
7 | df-br 4805 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) | |
8 | 2 | elexi 3353 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 5648 | . . . . 5 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) |
10 | 7, 9 | bitr4i 267 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 𝑏 ∈ ((t+‘𝑅) “ {𝑋})) |
11 | df-br 4805 | . . . . . 6 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) | |
12 | 8, 4 | elimasn 5648 | . . . . . 6 ⊢ (𝑎 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) |
13 | 11, 12 | bitr4i 267 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 𝑎 ∈ ((t+‘𝑅) “ {𝑋})) |
14 | 13 | imbi2i 325 | . . . 4 ⊢ ((𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎) ↔ (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 280 | . . 3 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
16 | 15 | alrimiv 2004 | . 2 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
17 | 1, 16 | mpg 1873 | 1 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1630 ∈ 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: frege98 38757 |
Copyright terms: Public domain | W3C validator |