Theorem latasym 18400

Description: A lattice ordering is asymmetric. (eqss 3938 analog.) (Contributed by NM, 8-Oct-2011.)

Hypotheses

Ref	Expression
latref.b	⊢ 𝐵 = (Base‘𝐾)
latref.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
latasym	⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))

Proof of Theorem latasym

Step	Hyp	Ref	Expression
1		latref.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		latref.l	. . 3 ⊢ ≤ = (le‘𝐾)
3	1, 2	latasymb 18399	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
4	3	biimpd 229	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  Basecbs 17170  lecple 17218  Latclat 18388

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-dm 5634  df-iota 6448  df-fv 6500  df-proset 18251  df-poset 18270  df-lat 18389

This theorem is referenced by: (None)