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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 43417 | . 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 3477 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3477 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | frege109.r | . . . . 5 ⊢ 𝑅 ∈ 𝑉 | |
6 | 2, 3, 4, 5 | frege108 43450 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 → (𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧)) |
7 | df-br 5153 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ ((t+‘𝑅) ∪ I )) | |
8 | 2 | elexi 3493 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 6098 | . . . . 5 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ ⟨𝑋, 𝑦⟩ ∈ ((t+‘𝑅) ∪ I )) |
10 | 7, 9 | bitr4i 277 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ 𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) |
11 | df-br 5153 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ ⟨𝑋, 𝑧⟩ ∈ ((t+‘𝑅) ∪ I )) | |
12 | 8, 4 | elimasn 6098 | . . . . . 6 ⊢ (𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ ⟨𝑋, 𝑧⟩ ∈ ((t+‘𝑅) ∪ I )) |
13 | 11, 12 | bitr4i 277 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) |
14 | 13 | imbi2i 335 | . . . 4 ⊢ ((𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧) ↔ (𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
15 | 6, 10, 14 | 3imtr3i 290 | . . 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 3473 ∪ cun 3947 {csn 4632 ⟨cop 4638 class class class wbr 5152 I cid 5579 “ cima 5685 ‘cfv 6553 t+ctcl 14974 hereditary whe 43251 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-frege1 43269 ax-frege2 43270 ax-frege8 43288 ax-frege28 43309 ax-frege31 43313 ax-frege41 43324 ax-frege52a 43336 ax-frege52c 43367 ax-frege58b 43380 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-seq 14009 df-trcl 14976 df-relexp 15009 df-he 43252 |
This theorem is referenced by: frege110 43452 |
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