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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 4098 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 18377 | . . 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 lecple 17304 ltcplt 18354 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-plt 18375 |
| This theorem is referenced by: pleval2 18382 pltnlt 18385 pltn2lp 18386 plttr 18387 pospo 18390 ogrpaddlt 33094 isarchi3 33194 archirngz 33196 archiabllem2a 33201 orngsqr 33334 ornglmullt 33337 orngrmullt 33338 atnlt 39314 cvlcvr1 39340 hlrelat 39404 hlrelat3 39414 cvratlem 39423 atltcvr 39437 atlelt 39440 llnnlt 39525 lplnnle2at 39543 lplnnlt 39567 lvolnle3at 39584 lvolnltN 39620 cdlemblem 39795 cdlemb 39796 lhpexle1 40010 |
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