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Mirrors > Home > MPE Home > Th. List > latasymb | Structured version Visualization version GIF version |
Description: A lattice ordering is asymmetric. (eqss 3836 analog.) (Contributed by NM, 22-Oct-2011.) |
Ref | Expression |
---|---|
latref.b | ⊢ 𝐵 = (Base‘𝐾) |
latref.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
latasymb | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latpos 17436 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
2 | latref.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | latref.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | posasymb 17338 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
5 | 1, 4 | syl3an1 1163 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 ‘cfv 6135 Basecbs 16255 lecple 16345 Posetcpo 17326 Latclat 17431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-dm 5365 df-iota 6099 df-fv 6143 df-proset 17314 df-poset 17332 df-lat 17432 |
This theorem is referenced by: latasym 17441 latasymd 17443 lubun 17509 cmtbr4N 35409 cvlexchb1 35484 hlateq 35553 cvratlem 35575 cvrat3 35596 pmap11 35916 cdleme50eq 36695 dia11N 37202 dib11N 37314 dih11 37419 |
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