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| Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version | ||
| Description: Deduce equality from lattice ordering. (eqssd 4001 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 18487 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) | 
| 9 | 3, 4, 5, 8 | syl3anc 1373 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Latclat 18476 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-proset 18340 df-poset 18359 df-lat 18477 | 
| This theorem is referenced by: latjidm 18507 latmidm 18519 latjass 18528 oldmm1 39218 olj01 39226 olm01 39237 cvlcvr1 39340 llnmlplnN 39541 2llnjaN 39568 2lplnja 39621 cdlema1N 39793 hlmod1i 39858 lautj 40095 lautm 40096 cdleme19a 40305 cdleme28b 40373 trljco 40742 dochvalr 41359 | 
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