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Mirrors > Home > MPE Home > Th. List > ple1 | Structured version Visualization version GIF version |
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
ple1.b | β’ π΅ = (BaseβπΎ) |
ple1.u | β’ π = (lubβπΎ) |
ple1.l | β’ β€ = (leβπΎ) |
ple1.1 | β’ 1 = (1.βπΎ) |
ple1.k | β’ (π β πΎ β π) |
ple1.x | β’ (π β π β π΅) |
ple1.d | β’ (π β π΅ β dom π) |
Ref | Expression |
---|---|
ple1 | β’ (π β π β€ 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ple1.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | ple1.l | . . 3 β’ β€ = (leβπΎ) | |
3 | ple1.u | . . 3 β’ π = (lubβπΎ) | |
4 | ple1.k | . . 3 β’ (π β πΎ β π) | |
5 | ple1.d | . . 3 β’ (π β π΅ β dom π) | |
6 | ple1.x | . . 3 β’ (π β π β π΅) | |
7 | 1, 2, 3, 4, 5, 6 | luble 18345 | . 2 β’ (π β π β€ (πβπ΅)) |
8 | ple1.1 | . . . 4 β’ 1 = (1.βπΎ) | |
9 | 1, 3, 8 | p1val 18414 | . . 3 β’ (πΎ β π β 1 = (πβπ΅)) |
10 | 4, 9 | syl 17 | . 2 β’ (π β 1 = (πβπ΅)) |
11 | 7, 10 | breqtrrd 5171 | 1 β’ (π β π β€ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 class class class wbr 5143 dom cdm 5673 βcfv 6543 Basecbs 17174 lecple 17234 lubclub 18295 1.cp1 18410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-lub 18332 df-p1 18412 |
This theorem is referenced by: ople1 38658 lhp2lt 39469 |
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