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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 4010 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 17838 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
4 | 3 | simprbda 502 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) |
5 | 4 | ex 416 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 class class class wbr 5053 ‘cfv 6380 lecple 16809 ltcplt 17815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-plt 17836 |
This theorem is referenced by: pleval2 17843 pltnlt 17846 pltn2lp 17847 plttr 17848 pospo 17851 ogrpaddlt 31062 isarchi3 31160 archirngz 31162 archiabllem2a 31167 orngsqr 31222 ornglmullt 31225 orngrmullt 31226 atnlt 37064 cvlcvr1 37090 hlrelat 37153 hlrelat3 37163 cvratlem 37172 atltcvr 37186 atlelt 37189 llnnlt 37274 lplnnle2at 37292 lplnnlt 37316 lvolnle3at 37333 lvolnltN 37369 cdlemblem 37544 cdlemb 37545 lhpexle1 37759 |
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