![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > latasym | Structured version Visualization version GIF version |
Description: A lattice ordering is asymmetric. (eqss 3992 analog.) (Contributed by NM, 8-Oct-2011.) |
Ref | Expression |
---|---|
latref.b | β’ π΅ = (BaseβπΎ) |
latref.l | β’ β€ = (leβπΎ) |
Ref | Expression |
---|---|
latasym | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latref.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | latref.l | . . 3 β’ β€ = (leβπΎ) | |
3 | 1, 2 | latasymb 18404 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
4 | 3 | biimpd 228 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 Basecbs 17150 lecple 17210 Latclat 18393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 df-iota 6488 df-fv 6544 df-proset 18257 df-poset 18275 df-lat 18394 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |