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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 43265 | . 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 3472 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3472 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | frege109.r | . . . . 5 ⊢ 𝑅 ∈ 𝑉 | |
6 | 2, 3, 4, 5 | frege108 43298 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 → (𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧)) |
7 | df-br 5142 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ ((t+‘𝑅) ∪ I )) | |
8 | 2 | elexi 3488 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 6082 | . . . . 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 5142 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ ⟨𝑋, 𝑧⟩ ∈ ((t+‘𝑅) ∪ I )) | |
12 | 8, 4 | elimasn 6082 | . . . . . 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 1922 | . 2 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) → ∀𝑧(𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
17 | 1, 16 | mpg 1791 | 1 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 {csn 4623 ⟨cop 4629 class class class wbr 5141 I cid 5566 “ cima 5672 ‘cfv 6537 t+ctcl 14938 hereditary whe 43099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 43117 ax-frege2 43118 ax-frege8 43136 ax-frege28 43157 ax-frege31 43161 ax-frege41 43172 ax-frege52a 43184 ax-frege52c 43215 ax-frege58b 43228 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 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 13973 df-trcl 14940 df-relexp 14973 df-he 43100 |
This theorem is referenced by: frege110 43300 |
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