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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 4013 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 18500 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
9 | 3, 4, 5, 8 | syl3anc 1370 | . 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Latclat 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-proset 18352 df-poset 18371 df-lat 18490 |
This theorem is referenced by: latjidm 18520 latmidm 18532 latjass 18541 oldmm1 39199 olj01 39207 olm01 39218 cvlcvr1 39321 llnmlplnN 39522 2llnjaN 39549 2lplnja 39602 cdlema1N 39774 hlmod1i 39839 lautj 40076 lautm 40077 cdleme19a 40286 cdleme28b 40354 trljco 40723 dochvalr 41340 |
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