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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege102 | Structured version Visualization version GIF version |
Description: If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege102.x | ⊢ 𝑋 ∈ 𝐴 |
frege102.z | ⊢ 𝑍 ∈ 𝐵 |
frege102.v | ⊢ 𝑉 ∈ 𝐶 |
frege102.r | ⊢ 𝑅 ∈ 𝐷 |
Ref | Expression |
---|---|
frege102 | ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege102.z | . . 3 ⊢ 𝑍 ∈ 𝐵 | |
2 | frege102.v | . . 3 ⊢ 𝑉 ∈ 𝐶 | |
3 | frege102.r | . . 3 ⊢ 𝑅 ∈ 𝐷 | |
4 | 1, 2, 3 | frege92 41533 | . 2 ⊢ (𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) |
5 | frege102.x | . . 3 ⊢ 𝑋 ∈ 𝐴 | |
6 | 5, 1, 2, 3 | frege96 41537 | . 2 ⊢ (𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) |
7 | 1 | frege101 41542 | . 2 ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)))) |
8 | 4, 6, 7 | mp2 9 | 1 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∪ cun 3886 class class class wbr 5076 I cid 5490 ‘cfv 6435 t+ctcl 14694 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-frege1 41368 ax-frege2 41369 ax-frege8 41387 ax-frege28 41408 ax-frege31 41412 ax-frege41 41423 ax-frege52a 41435 ax-frege52c 41466 ax-frege58b 41479 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-n0 12232 df-z 12318 df-uz 12581 df-seq 13720 df-trcl 14696 df-relexp 14729 df-he 41351 |
This theorem is referenced by: frege108 41549 |
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