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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege133d | Structured version Visualization version GIF version |
Description: If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 40340. (Contributed by RP, 18-Jul-2020.) |
Ref | Expression |
---|---|
frege133d.f | ⊢ (𝜑 → 𝐹 ∈ V) |
frege133d.xa | ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) |
frege133d.xb | ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) |
frege133d.fun | ⊢ (𝜑 → Fun 𝐹) |
Ref | Expression |
---|---|
frege133d | ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege133d.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) | |
2 | frege133d.xb | . . . . 5 ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) | |
3 | frege133d.fun | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) | |
4 | funrel 6371 | . . . . . . . 8 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → Rel 𝐹) |
6 | reltrclfv 14376 | . . . . . . 7 ⊢ ((𝐹 ∈ V ∧ Rel 𝐹) → Rel (t+‘𝐹)) | |
7 | 1, 5, 6 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → Rel (t+‘𝐹)) |
8 | eliniseg2 5968 | . . . . . 6 ⊢ (Rel (t+‘𝐹) → (𝑋 ∈ (◡(t+‘𝐹) “ {𝐵}) ↔ 𝑋(t+‘𝐹)𝐵)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (◡(t+‘𝐹) “ {𝐵}) ↔ 𝑋(t+‘𝐹)𝐵)) |
10 | 2, 9 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (◡(t+‘𝐹) “ {𝐵})) |
11 | frege133d.xa | . . . . 5 ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) | |
12 | brrelex2 5605 | . . . . 5 ⊢ ((Rel (t+‘𝐹) ∧ 𝑋(t+‘𝐹)𝐴) → 𝐴 ∈ V) | |
13 | 7, 11, 12 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | un12 4142 | . . . . . 6 ⊢ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) = ({𝐵} ∪ ((◡(t+‘𝐹) “ {𝐵}) ∪ ((t+‘𝐹) “ {𝐵}))) | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) = ({𝐵} ∪ ((◡(t+‘𝐹) “ {𝐵}) ∪ ((t+‘𝐹) “ {𝐵})))) |
16 | 1, 15, 3 | frege131d 40107 | . . . 4 ⊢ (𝜑 → (𝐹 “ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) ⊆ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
17 | 1, 10, 13, 11, 16 | frege83d 40091 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
18 | elun 4124 | . . . . 5 ⊢ (𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})) ↔ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) | |
19 | 18 | orbi2i 909 | . . . 4 ⊢ ((𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})))) |
20 | elun 4124 | . . . 4 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) | |
21 | 3orass 1086 | . . . 4 ⊢ ((𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})))) | |
22 | 19, 20, 21 | 3bitr4i 305 | . . 3 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
23 | 17, 22 | sylib 220 | . 2 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
24 | eliniseg2 5968 | . . . . 5 ⊢ (Rel (t+‘𝐹) → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ↔ 𝐴(t+‘𝐹)𝐵)) | |
25 | 7, 24 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ↔ 𝐴(t+‘𝐹)𝐵)) |
26 | 25 | biimpd 231 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) → 𝐴(t+‘𝐹)𝐵)) |
27 | elsni 4583 | . . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)) |
29 | elrelimasn 5952 | . . . . 5 ⊢ (Rel (t+‘𝐹) → (𝐴 ∈ ((t+‘𝐹) “ {𝐵}) ↔ 𝐵(t+‘𝐹)𝐴)) | |
30 | 7, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ((t+‘𝐹) “ {𝐵}) ↔ 𝐵(t+‘𝐹)𝐴)) |
31 | 30 | biimpd 231 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((t+‘𝐹) “ {𝐵}) → 𝐵(t+‘𝐹)𝐴)) |
32 | 26, 28, 31 | 3orim123d 1440 | . 2 ⊢ (𝜑 → ((𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})) → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴))) |
33 | 23, 32 | mpd 15 | 1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 {csn 4566 class class class wbr 5065 ◡ccnv 5553 “ cima 5557 Rel wrel 5559 Fun wfun 6348 ‘cfv 6354 t+ctcl 14344 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369 df-trcl 14346 df-relexp 14379 |
This theorem is referenced by: (None) |
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