![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege83 | Structured version Visualization version GIF version |
Description: Apply commuted form of frege81 40645 when the property 𝑅 is hereditary in a disjunction of two properties, only one of which is known to be held by 𝑋. Proposition 83 of [Frege1879] p. 65. Here we introduce the union of classes where Frege has a disjunction of properties which are represented by membership in either of the classes. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege83.x | ⊢ 𝑋 ∈ 𝑆 |
frege83.y | ⊢ 𝑌 ∈ 𝑇 |
frege83.r | ⊢ 𝑅 ∈ 𝑈 |
frege83.b | ⊢ 𝐵 ∈ 𝑉 |
frege83.c | ⊢ 𝐶 ∈ 𝑊 |
Ref | Expression |
---|---|
frege83 | ⊢ (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege36 40540 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
2 | elun 4076 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
3 | df-or 845 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
4 | 2, 3 | bitri 278 | . . 3 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) |
5 | 1, 4 | sylibr 237 | . 2 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (𝐵 ∪ 𝐶)) |
6 | frege83.x | . . 3 ⊢ 𝑋 ∈ 𝑆 | |
7 | frege83.y | . . 3 ⊢ 𝑌 ∈ 𝑇 | |
8 | frege83.r | . . 3 ⊢ 𝑅 ∈ 𝑈 | |
9 | frege83.b | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
10 | 9 | elexi 3460 | . . . 4 ⊢ 𝐵 ∈ V |
11 | frege83.c | . . . . 5 ⊢ 𝐶 ∈ 𝑊 | |
12 | 11 | elexi 3460 | . . . 4 ⊢ 𝐶 ∈ V |
13 | 10, 12 | unex 7449 | . . 3 ⊢ (𝐵 ∪ 𝐶) ∈ V |
14 | 6, 7, 8, 13 | frege82 40646 | . 2 ⊢ ((𝑋 ∈ 𝐵 → 𝑋 ∈ (𝐵 ∪ 𝐶)) → (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶))))) |
15 | 5, 14 | ax-mp 5 | 1 ⊢ (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 class class class wbr 5030 ‘cfv 6324 t+ctcl 14336 hereditary whe 40473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-frege1 40491 ax-frege2 40492 ax-frege8 40510 ax-frege28 40531 ax-frege31 40535 ax-frege52a 40558 ax-frege58b 40602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-trcl 14338 df-relexp 14371 df-he 40474 |
This theorem is referenced by: frege133 40697 |
Copyright terms: Public domain | W3C validator |