Theorem posasymb 18225

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 1136	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Poset)
2		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
3		simp3 1138	. . . 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 18223	. . . 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 1374	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌) → 𝑋 ≤ 𝑌)))
8	7	simp2d 1143	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
9	4, 5	posref 18224	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
10		breq2 5096	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
11	9, 10	syl5ibcom 245	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
12		breq1 5095	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
13	9, 12	syl5ibcom 245	. . . 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 1132	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋)))
16	8, 15	impbid 212	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5092  ‘cfv 6482  Basecbs 17120  lecple 17168  Posetcpo 18213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-proset 18200  df-poset 18219

This theorem is referenced by:  odupos  18232  pltnle  18242  pltval3  18243  lublecllem  18264  poslubmo  18315  posglbmo  18316  latasymb  18348  latleeqj1  18357  latleeqm1  18373  posrasymb  32909  mgcf1olem1  32943  mgcf1olem2  32944  archirngz  33131  archiabllem1a  33133  ople0  39166  op1le  39171  atlle0  39284