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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 40162 | . 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 3495 | . . . . 5 ⊢ 𝑏 ∈ V | |
4 | vex 3495 | . . . . 5 ⊢ 𝑎 ∈ V | |
5 | frege97.r | . . . . 5 ⊢ 𝑅 ∈ 𝑊 | |
6 | 2, 3, 4, 5 | frege96 40183 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 → (𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎)) |
7 | df-br 5058 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) | |
8 | 2 | elexi 3511 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 5947 | . . . . 5 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) |
10 | 7, 9 | bitr4i 279 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 𝑏 ∈ ((t+‘𝑅) “ {𝑋})) |
11 | df-br 5058 | . . . . . 6 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) | |
12 | 8, 4 | elimasn 5947 | . . . . . 6 ⊢ (𝑎 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) |
13 | 11, 12 | bitr4i 279 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 𝑎 ∈ ((t+‘𝑅) “ {𝑋})) |
14 | 13 | imbi2i 337 | . . . 4 ⊢ ((𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎) ↔ (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 292 | . . 3 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
16 | 15 | alrimiv 1919 | . 2 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
17 | 1, 16 | mpg 1789 | 1 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 class class class wbr 5057 “ cima 5551 ‘cfv 6348 t+ctcl 14333 hereditary whe 39996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-frege1 40014 ax-frege2 40015 ax-frege8 40033 ax-frege52a 40081 ax-frege58b 40125 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-trcl 14335 df-relexp 14368 df-he 39997 |
This theorem is referenced by: frege98 40185 |
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