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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 18253 | . 2 β’ (π β π β€ (πβπ΅)) |
8 | ple1.1 | . . . 4 β’ 1 = (1.βπΎ) | |
9 | 1, 3, 8 | p1val 18322 | . . 3 β’ (πΎ β π β 1 = (πβπ΅)) |
10 | 4, 9 | syl 17 | . 2 β’ (π β 1 = (πβπ΅)) |
11 | 7, 10 | breqtrrd 5134 | 1 β’ (π β π β€ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5106 dom cdm 5634 βcfv 6497 Basecbs 17088 lecple 17145 lubclub 18203 1.cp1 18318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-lub 18240 df-p1 18320 |
This theorem is referenced by: ople1 37699 lhp2lt 38510 |
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