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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 42560 | . 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 3479 | . . . . 5 ⊢ 𝑏 ∈ V | |
4 | vex 3479 | . . . . 5 ⊢ 𝑎 ∈ V | |
5 | frege97.r | . . . . 5 ⊢ 𝑅 ∈ 𝑊 | |
6 | 2, 3, 4, 5 | frege96 42581 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 → (𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎)) |
7 | df-br 5145 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) | |
8 | 2 | elexi 3494 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 6080 | . . . . 5 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑏〉 ∈ (t+‘𝑅)) |
10 | 7, 9 | bitr4i 278 | . . . 4 ⊢ (𝑋(t+‘𝑅)𝑏 ↔ 𝑏 ∈ ((t+‘𝑅) “ {𝑋})) |
11 | df-br 5145 | . . . . . 6 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) | |
12 | 8, 4 | elimasn 6080 | . . . . . 6 ⊢ (𝑎 ∈ ((t+‘𝑅) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ (t+‘𝑅)) |
13 | 11, 12 | bitr4i 278 | . . . . 5 ⊢ (𝑋(t+‘𝑅)𝑎 ↔ 𝑎 ∈ ((t+‘𝑅) “ {𝑋})) |
14 | 13 | imbi2i 336 | . . . 4 ⊢ ((𝑏𝑅𝑎 → 𝑋(t+‘𝑅)𝑎) ↔ (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 291 | . . 3 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → (𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
16 | 15 | alrimiv 1931 | . 2 ⊢ (𝑏 ∈ ((t+‘𝑅) “ {𝑋}) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((t+‘𝑅) “ {𝑋}))) |
17 | 1, 16 | mpg 1800 | 1 ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∈ wcel 2107 Vcvv 3475 {csn 4624 〈cop 4630 class class class wbr 5144 “ cima 5675 ‘cfv 6535 t+ctcl 14919 hereditary whe 42394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-frege1 42412 ax-frege2 42413 ax-frege8 42431 ax-frege52a 42479 ax-frege58b 42523 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-n0 12460 df-z 12546 df-uz 12810 df-seq 13954 df-trcl 14921 df-relexp 14954 df-he 42395 |
This theorem is referenced by: frege98 42583 |
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