![]() |
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 39079 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 38974 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
2 | elun 3981 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
3 | df-or 881 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
4 | 2, 3 | bitri 267 | . . 3 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) |
5 | 1, 4 | sylibr 226 | . 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 3431 | . . . 4 ⊢ 𝐵 ∈ V |
11 | frege83.c | . . . . 5 ⊢ 𝐶 ∈ 𝑊 | |
12 | 11 | elexi 3431 | . . . 4 ⊢ 𝐶 ∈ V |
13 | 10, 12 | unex 7217 | . . 3 ⊢ (𝐵 ∪ 𝐶) ∈ V |
14 | 6, 7, 8, 13 | frege82 39080 | . 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 880 ∈ wcel 2166 Vcvv 3415 ∪ cun 3797 class class class wbr 4874 ‘cfv 6124 t+ctcl 14104 hereditary whe 38907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-frege1 38925 ax-frege2 38926 ax-frege8 38944 ax-frege28 38965 ax-frege31 38969 ax-frege52a 38992 ax-frege58b 39036 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-n0 11620 df-z 11706 df-uz 11970 df-seq 13097 df-trcl 14106 df-relexp 14139 df-he 38908 |
This theorem is referenced by: frege133 39131 |
Copyright terms: Public domain | W3C validator |