![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege126d | Structured version Visualization version GIF version |
Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 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. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 39123. (Contributed by RP, 16-Jul-2020.) |
Ref | Expression |
---|---|
frege126d.f | ⊢ (𝜑 → 𝐹 ∈ V) |
frege126d.x | ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
frege126d.a | ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) |
frege126d.xb | ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) |
frege126d.fun | ⊢ (𝜑 → Fun 𝐹) |
Ref | Expression |
---|---|
frege126d | ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege126d.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) | |
2 | frege126d.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) | |
3 | frege126d.a | . . 3 ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) | |
4 | frege126d.xb | . . 3 ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) | |
5 | frege126d.fun | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
6 | 1, 2, 3, 4, 5 | frege124d 38894 | . 2 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) |
7 | 6 | frege114d 38891 | 1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1112 = wceq 1658 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 dom cdm 5342 Fun wfun 6117 ‘cfv 6123 t+ctcl 14103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-seq 13096 df-trcl 14105 df-relexp 14138 |
This theorem is referenced by: frege129d 38896 |
Copyright terms: Public domain | W3C validator |