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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnlen0 | Structured version Visualization version GIF version |
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 3030 and op0le 36316 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.) |
Ref | Expression |
---|---|
op0le.b | ⊢ 𝐵 = (Base‘𝐾) |
op0le.l | ⊢ ≤ = (le‘𝐾) |
op0le.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
opnlen0 | ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑋 ≤ 𝑌) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op0le.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | op0le.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
3 | op0le.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | op0le 36316 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → 0 ≤ 𝑌) |
5 | 4 | 3adant2 1127 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 0 ≤ 𝑌) |
6 | breq1 5062 | . . . 4 ⊢ (𝑋 = 0 → (𝑋 ≤ 𝑌 ↔ 0 ≤ 𝑌)) | |
7 | 5, 6 | syl5ibrcom 249 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 0 → 𝑋 ≤ 𝑌)) |
8 | 7 | necon3bd 3030 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 → 𝑋 ≠ 0 )) |
9 | 8 | imp 409 | 1 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑋 ≤ 𝑌) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 lecple 16566 0.cp0 17641 OPcops 36302 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 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-iun 4914 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-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-glb 17579 df-p0 17643 df-oposet 36306 |
This theorem is referenced by: cdlemg12e 37777 |
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