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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3904 analog.) (Contributed by NM, 18-Nov-2011.) |
Ref | Expression |
---|---|
latasymd.b | ⊢ 𝐵 = (Base‘𝐾) |
latasymd.l | ⊢ ≤ = (le‘𝐾) |
latasymd.3 | ⊢ (𝜑 → 𝐾 ∈ Lat) |
latasymd.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
latasymd.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
latasymd.6 | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
latasymd.7 | ⊢ (𝜑 → 𝑌 ≤ 𝑋) |
Ref | Expression |
---|---|
latasymd | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latasymd.6 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
2 | latasymd.7 | . 2 ⊢ (𝜑 → 𝑌 ≤ 𝑋) | |
3 | latasymd.3 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Lat) | |
4 | latasymd.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | latasymd.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | latasymd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
7 | latasymd.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
8 | 6, 7 | latasymb 17793 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
9 | 3, 4, 5, 8 | syl3anc 1372 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
10 | 1, 2, 9 | mpbi2and 712 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 Basecbs 16599 lecple 16688 Latclat 17784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-xp 5541 df-dm 5545 df-iota 6308 df-fv 6358 df-proset 17667 df-poset 17685 df-lat 17785 |
This theorem is referenced by: latjidm 17813 latmidm 17825 latjass 17834 oldmm1 36887 olj01 36895 olm01 36906 cvlcvr1 37009 llnmlplnN 37209 2llnjaN 37236 2lplnja 37289 cdlema1N 37461 hlmod1i 37526 lautj 37763 lautm 37764 cdleme19a 37973 cdleme28b 38041 trljco 38410 dochvalr 39027 |
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