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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege83 | Structured version Visualization version GIF version |
Description: Apply commuted form of frege81 43377 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 43272 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
2 | elun 4147 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
3 | df-or 846 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
4 | 2, 3 | bitri 274 | . . 3 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) |
5 | 1, 4 | sylibr 233 | . 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 3491 | . . . 4 ⊢ 𝐵 ∈ V |
11 | frege83.c | . . . . 5 ⊢ 𝐶 ∈ 𝑊 | |
12 | 11 | elexi 3491 | . . . 4 ⊢ 𝐶 ∈ V |
13 | 10, 12 | unex 7752 | . . 3 ⊢ (𝐵 ∪ 𝐶) ∈ V |
14 | 6, 7, 8, 13 | frege82 43378 | . 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 845 ∈ wcel 2098 Vcvv 3471 ∪ cun 3945 class class class wbr 5150 ‘cfv 6551 t+ctcl 14970 hereditary whe 43205 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-frege1 43223 ax-frege2 43224 ax-frege8 43242 ax-frege28 43263 ax-frege31 43267 ax-frege52a 43290 ax-frege58b 43334 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-n0 12509 df-z 12595 df-uz 12859 df-seq 14005 df-trcl 14972 df-relexp 15005 df-he 43206 |
This theorem is referenced by: frege133 43429 |
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