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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 4069 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 17558 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
4 | 3 | simprbda 499 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) |
5 | 4 | ex 413 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 lecple 16560 ltcplt 17539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-plt 17556 |
This theorem is referenced by: pleval2 17563 pltnlt 17566 pltn2lp 17567 plttr 17568 pospo 17571 ogrpaddlt 30645 isarchi3 30743 archirngz 30745 archiabllem2a 30750 orngsqr 30804 ornglmullt 30807 orngrmullt 30808 atnlt 36329 cvlcvr1 36355 hlrelat 36418 hlrelat3 36428 cvratlem 36437 atltcvr 36451 atlelt 36454 llnnlt 36539 lplnnle2at 36557 lplnnlt 36581 lvolnle3at 36598 lvolnltN 36634 cdlemblem 36809 cdlemb 36810 lhpexle1 37024 |
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