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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 40213. (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 39984 | . 2 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) |
7 | 6 | frege114d 39981 | 1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 t+ctcl 14333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 df-trcl 14335 df-relexp 14368 |
This theorem is referenced by: frege129d 39986 |
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