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Mirrors > Home > MPE Home > Th. List > latasym | Structured version Visualization version GIF version |
Description: A lattice ordering is asymmetric. (eqss 3997 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 18441 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 Latclat 18430 |
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 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-dm 5692 df-iota 6505 df-fv 6561 df-proset 18294 df-poset 18312 df-lat 18431 |
This theorem is referenced by: (None) |
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