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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3996 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 18379 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
9 | 3, 4, 5, 8 | syl3anc 1371 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
10 | 1, 2, 9 | mpbi2and 710 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5142 ‘cfv 6533 Basecbs 17128 lecple 17188 Latclat 18368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-xp 5676 df-dm 5680 df-iota 6485 df-fv 6541 df-proset 18232 df-poset 18250 df-lat 18369 |
This theorem is referenced by: latjidm 18399 latmidm 18411 latjass 18420 oldmm1 37956 olj01 37964 olm01 37975 cvlcvr1 38078 llnmlplnN 38279 2llnjaN 38306 2lplnja 38359 cdlema1N 38531 hlmod1i 38596 lautj 38833 lautm 38834 cdleme19a 39043 cdleme28b 39111 trljco 39480 dochvalr 40097 |
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