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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 40697. (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 6341 | . . . . . . . 8 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → Rel 𝐹) |
6 | reltrclfv 14368 | . . . . . . 7 ⊢ ((𝐹 ∈ V ∧ Rel 𝐹) → Rel (t+‘𝐹)) | |
7 | 1, 5, 6 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → Rel (t+‘𝐹)) |
8 | eliniseg2 5936 | . . . . . 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 260 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (◡(t+‘𝐹) “ {𝐵})) |
11 | frege133d.xa | . . . . 5 ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) | |
12 | brrelex2 5570 | . . . . 5 ⊢ ((Rel (t+‘𝐹) ∧ 𝑋(t+‘𝐹)𝐴) → 𝐴 ∈ V) | |
13 | 7, 11, 12 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | un12 4094 | . . . . . 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 40465 | . . . 4 ⊢ (𝜑 → (𝐹 “ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) ⊆ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
17 | 1, 10, 13, 11, 16 | frege83d 40449 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) |
18 | elun 4076 | . . . . 5 ⊢ (𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})) ↔ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) | |
19 | 18 | orbi2i 910 | . . . 4 ⊢ ((𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ (𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵})))) |
20 | elun 4076 | . . . 4 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵})))) | |
21 | 3orass 1087 | . . . 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 306 | . . 3 ⊢ (𝐴 ∈ ((◡(t+‘𝐹) “ {𝐵}) ∪ ({𝐵} ∪ ((t+‘𝐹) “ {𝐵}))) ↔ (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
23 | 17, 22 | sylib 221 | . 2 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) ∨ 𝐴 ∈ {𝐵} ∨ 𝐴 ∈ ((t+‘𝐹) “ {𝐵}))) |
24 | eliniseg2 5936 | . . . . 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 232 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (◡(t+‘𝐹) “ {𝐵}) → 𝐴(t+‘𝐹)𝐵)) |
27 | elsni 4542 | . . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)) |
29 | elrelimasn 5920 | . . . . 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 232 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((t+‘𝐹) “ {𝐵}) → 𝐵(t+‘𝐹)𝐴)) |
32 | 26, 28, 31 | 3orim123d 1441 | . 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 209 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 {csn 4525 class class class wbr 5030 ◡ccnv 5518 “ cima 5522 Rel wrel 5524 Fun wfun 6318 ‘cfv 6324 t+ctcl 14336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-trcl 14338 df-relexp 14371 |
This theorem is referenced by: (None) |
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