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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3585 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 16877 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
9 | 3, 4, 5, 8 | syl3anc 1318 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
10 | 1, 2, 9 | mpbi2and 958 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-preset 16751 df-poset 16769 df-lat 16869 |
This theorem is referenced by: latjidm 16897 latmidm 16909 latjass 16918 oldmm1 33522 olj01 33530 olm01 33541 cvlcvr1 33644 llnmlplnN 33843 2llnjaN 33870 2lplnja 33923 cdlema1N 34095 hlmod1i 34160 lautj 34397 lautm 34398 cdleme19a 34609 cdleme28b 34677 trljco 35046 dochvalr 35664 |
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