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Mirrors > Home > MPE Home > Th. List > pltle | Structured version Visualization version GIF version |
Description: "Less than" implies "less than or equal to". (pssss 4108 analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
pltval.l | ⊢ ≤ = (le‘𝐾) |
pltval.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltle | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | pltval.s | . . . 4 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltval 18390 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
4 | 3 | simprbda 498 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) |
5 | 4 | ex 412 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 lecple 17305 ltcplt 18366 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-plt 18388 |
This theorem is referenced by: pleval2 18395 pltnlt 18398 pltn2lp 18399 plttr 18400 pospo 18403 ogrpaddlt 33077 isarchi3 33177 archirngz 33179 archiabllem2a 33184 orngsqr 33314 ornglmullt 33317 orngrmullt 33318 atnlt 39295 cvlcvr1 39321 hlrelat 39385 hlrelat3 39395 cvratlem 39404 atltcvr 39418 atlelt 39421 llnnlt 39506 lplnnle2at 39524 lplnnlt 39548 lvolnle3at 39565 lvolnltN 39601 cdlemblem 39776 cdlemb 39777 lhpexle1 39991 |
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