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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 42991 | . 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 3478 | . . . . 5 ⊢ 𝑏 ∈ V | |
4 | vex 3478 | . . . . 5 ⊢ 𝑎 ∈ V | |
5 | frege97.r | . . . . 5 ⊢ 𝑅 ∈ 𝑊 | |
6 | 2, 3, 4, 5 | frege96 43012 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 → (𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎)) |
7 | df-br 5149 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ ⟨𝑋, 𝑏⟩ ∈ (t+‘𝑅)) | |
8 | 2 | elexi 3493 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 6088 | . . . . 5 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) ↔ ⟨𝑋, 𝑏⟩ ∈ (t+‘𝑅)) |
10 | 7, 9 | bitr4i 277 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 𝑏 ∈ ((t+‘𝑅) “ {𝑋})) |
11 | df-br 5149 | . . . . . 6 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ ⟨𝑋, 𝑎⟩ ∈ (t+‘𝑅)) | |
12 | 8, 4 | elimasn 6088 | . . . . . 6 ⊢ (𝑎 ∈ ((t+‘𝑅) “ {𝑋}) ↔ ⟨𝑋, 𝑎⟩ ∈ (t+‘𝑅)) |
13 | 11, 12 | bitr4i 277 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 𝑎 ∈ ((t+‘𝑅) “ {𝑋})) |
14 | 13 | imbi2i 335 | . . . 4 ⊢ ((𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎) ↔ (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 290 | . . 3 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
16 | 15 | alrimiv 1930 | . 2 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
17 | 1, 16 | mpg 1799 | 1 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2106 Vcvv 3474 {csn 4628 ⟨cop 4634 class class class wbr 5148 “ cima 5679 ‘cfv 6543 t+ctcl 14936 hereditary whe 42825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-frege1 42843 ax-frege2 42844 ax-frege8 42862 ax-frege52a 42910 ax-frege58b 42954 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-seq 13971 df-trcl 14938 df-relexp 14971 df-he 42826 |
This theorem is referenced by: frege98 43014 |
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