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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 41499 | . 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 3434 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3434 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | frege109.r | . . . . 5 ⊢ 𝑅 ∈ 𝑉 | |
6 | 2, 3, 4, 5 | frege108 41532 | . . . 4 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 → (𝑦𝑅𝑧 → 𝑋((t+‘𝑅) ∪ I )𝑧)) |
7 | df-br 5079 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑦 ↔ 〈𝑋, 𝑦〉 ∈ ((t+‘𝑅) ∪ I )) | |
8 | 2 | elexi 3449 | . . . . . 6 ⊢ 𝑋 ∈ V |
9 | 8, 3 | elimasn 5994 | . . . . 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 5079 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑧 ↔ 〈𝑋, 𝑧〉 ∈ ((t+‘𝑅) ∪ I )) | |
12 | 8, 4 | elimasn 5994 | . . . . . 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 1933 | . 2 ⊢ (𝑦 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) → ∀𝑧(𝑦𝑅𝑧 → 𝑧 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}))) |
17 | 1, 16 | mpg 1803 | 1 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2109 Vcvv 3430 ∪ cun 3889 {csn 4566 〈cop 4572 class class class wbr 5078 I cid 5487 “ cima 5591 ‘cfv 6430 t+ctcl 14677 hereditary whe 41333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-frege1 41351 ax-frege2 41352 ax-frege8 41370 ax-frege28 41391 ax-frege31 41395 ax-frege41 41406 ax-frege52a 41418 ax-frege52c 41449 ax-frege58b 41462 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-n0 12217 df-z 12303 df-uz 12565 df-seq 13703 df-trcl 14679 df-relexp 14712 df-he 41334 |
This theorem is referenced by: frege110 41534 |
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