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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege83 | Structured version Visualization version GIF version |
Description: Apply commuted form of frege81 40283 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 40178 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
2 | elun 4124 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
3 | df-or 844 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐶)) |
5 | 1, 4 | sylibr 236 | . 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 3513 | . . . 4 ⊢ 𝐵 ∈ V |
11 | frege83.c | . . . . 5 ⊢ 𝐶 ∈ 𝑊 | |
12 | 11 | elexi 3513 | . . . 4 ⊢ 𝐶 ∈ V |
13 | 10, 12 | unex 7463 | . . 3 ⊢ (𝐵 ∪ 𝐶) ∈ V |
14 | 6, 7, 8, 13 | frege82 40284 | . 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 843 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 class class class wbr 5058 ‘cfv 6349 t+ctcl 14339 hereditary whe 40111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-frege1 40129 ax-frege2 40130 ax-frege8 40148 ax-frege28 40169 ax-frege31 40173 ax-frege52a 40196 ax-frege58b 40240 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-trcl 14341 df-relexp 14374 df-he 40112 |
This theorem is referenced by: frege133 40335 |
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