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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrlt | Structured version Visualization version GIF version |
Description: The covers relation implies the less-than relation. (cvpss 30056 analog.) (Contributed by NM, 8-Oct-2011.) |
Ref | Expression |
---|---|
cvrfval.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrfval.s | ⊢ < = (lt‘𝐾) |
cvrfval.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
cvrlt | ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cvrfval.s | . . 3 ⊢ < = (lt‘𝐾) | |
3 | cvrfval.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrval 36399 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)))) |
5 | 4 | simprbda 501 | 1 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 ltcplt 17545 ⋖ ccvr 36392 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-covers 36396 |
This theorem is referenced by: ncvr1 36402 cvrletrN 36403 cvrnbtwn2 36405 cvrnbtwn3 36406 cvrle 36408 cvrnle 36410 cvrne 36411 0ltat 36421 atlen0 36440 atcvreq0 36444 cvlcvr1 36469 cvrval3 36543 cvrval4N 36544 cvrexchlem 36549 ltcvrntr 36554 cvrntr 36555 cvrat2 36559 atltcvr 36565 1cvratex 36603 ps-2 36608 llnnleat 36643 lplnnle2at 36671 lvolnle3at 36712 lhp0lt 37133 |
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