![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 38677. (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 5986 | . . . . . . . 8 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → Rel 𝐹) |
6 | reltrclfv 13846 | . . . . . . 7 ⊢ ((𝐹 ∈ V ∧ Rel 𝐹) → Rel (t+‘𝐹)) | |
7 | 1, 5, 6 | syl2anc 696 | . . . . . 6 ⊢ (𝜑 → Rel (t+‘𝐹)) |
8 | eliniseg2 5583 | . . . . . 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 247 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (◡(t+‘𝐹) “ {𝐵})) |
11 | frege133d.xa | . . . . 5 ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) | |
12 | brrelex2 5234 | . . . . 5 ⊢ ((Rel (t+‘𝐹) ∧ 𝑋(t+‘𝐹)𝐴) → 𝐴 ∈ V) | |
13 | 7, 11, 12 | syl2anc 696 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | un12 3847 | . . . . . 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 38443 | . . . 4 ⊢ (𝜑 → (𝐹 “ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) ⊆ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
17 | 1, 10, 13, 11, 16 | frege83d 38427 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
18 | elun 3829 | . . . . 5 ⊢ (𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})) ↔ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) | |
19 | 18 | orbi2i 542 | . . . 4 ⊢ ((𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})))) |
20 | elun 3829 | . . . 4 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) | |
21 | 3orass 1075 | . . . 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 292 | . . 3 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
23 | 17, 22 | sylib 208 | . 2 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
24 | eliniseg2 5583 | . . . . 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 219 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) → 𝐴(t+‘𝐹)𝐵)) |
27 | elsni 4270 | . . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)) |
29 | elrelimasn 5567 | . . . . 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 219 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((t+‘𝐹) “ {𝐵}) → 𝐵(t+‘𝐹)𝐴)) |
32 | 26, 28, 31 | 3orim123d 1488 | . 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 196 ∨ wo 382 ∨ w3o 1071 = wceq 1564 ∈ wcel 2071 Vcvv 3272 ∪ cun 3646 {csn 4253 class class class wbr 4728 ◡ccnv 5185 “ cima 5189 Rel wrel 5191 Fun wfun 5963 ‘cfv 5969 t+ctcl 13814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-er 7830 df-en 8041 df-dom 8042 df-sdom 8043 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-2 11160 df-n0 11374 df-z 11459 df-uz 11769 df-fz 12409 df-seq 12885 df-trcl 13816 df-relexp 13849 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |