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Mirrors > Home > MPE Home > Th. List > prsref | Structured version Visualization version GIF version |
Description: "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
isprs.b | β’ π΅ = (BaseβπΎ) |
isprs.l | β’ β€ = (leβπΎ) |
Ref | Expression |
---|---|
prsref | β’ ((πΎ β Proset β§ π β π΅) β π β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 β’ (π β π΅ β π β π΅) | |
2 | 1, 1, 1 | 3jca 1128 | . . 3 β’ (π β π΅ β (π β π΅ β§ π β π΅ β§ π β π΅)) |
3 | isprs.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
4 | isprs.l | . . . 4 β’ β€ = (leβπΎ) | |
5 | 3, 4 | prslem 18255 | . . 3 β’ ((πΎ β Proset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β€ π β§ ((π β€ π β§ π β€ π) β π β€ π))) |
6 | 2, 5 | sylan2 593 | . 2 β’ ((πΎ β Proset β§ π β π΅) β (π β€ π β§ ((π β€ π β§ π β€ π) β π β€ π))) |
7 | 6 | simpld 495 | 1 β’ ((πΎ β Proset β§ π β π΅) β π β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 Basecbs 17148 lecple 17208 Proset cproset 18250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-proset 18252 |
This theorem is referenced by: posref 18275 mgccole1 32415 mgccole2 32416 prsdm 33180 prsrn 33181 prsthinc 47762 |
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