Theorem posasymb 18296

Description: A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.)

Hypotheses

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

Assertion

Ref	Expression
posasymb	⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))

Proof of Theorem posasymb

Step	Hyp	Ref	Expression
1		simp1 1134	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Poset)
2		simp2 1135	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
3		simp3 1136	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
4		posi.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		posi.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6	4, 5	posi 18294	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌) → 𝑋 ≤ 𝑌)))
7	1, 2, 3, 3, 6	syl13anc 1370	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌) → 𝑋 ≤ 𝑌)))
8	7	simp2d 1141	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
9	4, 5	posref 18295	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
10		breq2 5146	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
11	9, 10	syl5ibcom 244	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
12		breq1 5145	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
13	9, 12	syl5ibcom 244	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋))
14	11, 13	jcad 512	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋)))
15	14	3adant3 1130	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋)))
16	8, 15	impbid 211	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  ‘cfv 6542  Basecbs 17165  lecple 17225  Posetcpo 18284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-proset 18272  df-poset 18290

This theorem is referenced by:  odupos  18305  pltnle  18315  pltval3  18316  lublecllem  18337  poslubmo  18388  posglbmo  18389  latasymb  18419  latleeqj1  18428  latleeqm1  18444  posrasymb  32661  mgcf1olem1  32697  mgcf1olem2  32698  archirngz  32862  archiabllem1a  32864  ople0  38583  op1le  38588  atlle0  38701