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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege109 | Structured version Visualization version GIF version |
Description: The property of belonging to the 𝑅-sequence beginning with 𝑋 is hereditary in the 𝑅-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege109.x | ⊢ 𝑋 ∈ 𝑈 |
frege109.r | ⊢ 𝑅 ∈ 𝑉 |
Ref | Expression |
---|---|
frege109 | ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege75 43359 | . 2 ⊢ (∀𝑦(𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) → ∀𝑧(𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) → 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})) | |
2 | frege109.x | . . . . 5 ⊢ 𝑋 ∈ 𝑈 | |
3 | vex 3474 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3474 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | frege109.r | . . . . 5 ⊢ 𝑅 ∈ 𝑉 | |
6 | 2, 3, 4, 5 | frege108 43392 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 → (𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧)) |
7 | df-br 5144 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ 〈𝑋, 𝑦〉 ∈ ((t+‘𝑅) ∪ I )) | |
8 | 2 | elexi 3490 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 6088 | . . . . 5 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑦〉 ∈ ((t+‘𝑅) ∪ I )) |
10 | 7, 9 | bitr4i 278 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ 𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) |
11 | df-br 5144 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ 〈𝑋, 𝑧〉 ∈ ((t+‘𝑅) ∪ I )) | |
12 | 8, 4 | elimasn 6088 | . . . . . 6 ⊢ (𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑧〉 ∈ ((t+‘𝑅) ∪ I )) |
13 | 11, 12 | bitr4i 278 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) |
14 | 13 | imbi2i 336 | . . . 4 ⊢ ((𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧) ↔ (𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 291 | . . 3 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) → (𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
16 | 15 | alrimiv 1923 | . 2 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) → ∀𝑧(𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
17 | 1, 16 | mpg 1792 | 1 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 ∈ wcel 2099 Vcvv 3470 ∪ cun 3943 {csn 4625 〈cop 4631 class class class wbr 5143 I cid 5570 “ cima 5676 ‘cfv 6543 t+ctcl 14959 hereditary whe 43193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-frege1 43211 ax-frege2 43212 ax-frege8 43230 ax-frege28 43251 ax-frege31 43255 ax-frege41 43266 ax-frege52a 43278 ax-frege52c 43309 ax-frege58b 43322 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-seq 13994 df-trcl 14961 df-relexp 14994 df-he 43194 |
This theorem is referenced by: frege110 43394 |
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