Theorem posrasymb 28988

Description: A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)

Hypotheses

Ref	Expression
posrasymb.b	⊢ 𝐵 = (Base‘𝐾)
posrasymb.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))

Assertion

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

Proof of Theorem posrasymb

Step	Hyp	Ref	Expression
1		posrasymb.l	. . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2	1	breqi 4589	. . . 4 ⊢ (𝑋 ≤ 𝑌 ↔ 𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
3		simp2 1055	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
4		simp3 1056	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
5		brxp 5071	. . . . . 6 ⊢ (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
6	3, 4, 5	sylanbrc 695	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7		brin 4634	. . . . . 6 ⊢ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋(𝐵 × 𝐵)𝑌))
8	7	rbaib 945	. . . . 5 ⊢ (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
9	6, 8	syl 17	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
10	2, 9	syl5bb 271	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐾)𝑌))
11	1	breqi 4589	. . . 4 ⊢ (𝑌 ≤ 𝑋 ↔ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
12		brxp 5071	. . . . . 6 ⊢ (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
13	4, 3, 12	sylanbrc 695	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
14		brin 4634	. . . . . 6 ⊢ (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋 ∧ 𝑌(𝐵 × 𝐵)𝑋))
15	14	rbaib 945	. . . . 5 ⊢ (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
16	13, 15	syl 17	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
17	11, 16	syl5bb 271	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ≤ 𝑋 ↔ 𝑌(le‘𝐾)𝑋))
18	10, 17	anbi12d 743	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋)))
19		posrasymb.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
20		eqid 2610	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
21	19, 20	posasymb 16775	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌))
22	18, 21	bitrd 267	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   class class class wbr 4583   × cxp 5036  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-iota 5768  df-fv 5812  df-preset 16751  df-poset 16769

This theorem is referenced by:  ordtconlem1  29298