Theorem posrasymb 33353

Description: A poset ordering is asymmetric. (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 5117	. . . 4 ⊢ (𝑋 ≤ 𝑌 ↔ 𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
3		simp2 1155	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
4		simp3 1156	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
5		brxp 5712	. . . . . 6 ⊢ (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
6	3, 4, 5	sylanbrc 595	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7		brin 5165	. . . . . 6 ⊢ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋(𝐵 × 𝐵)𝑌))
8	7	rbaib 548	. . . . 5 ⊢ (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
9	6, 8	syl 18	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
10	2, 9	bitrid 286	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐾)𝑌))
11	1	breqi 5117	. . . 4 ⊢ (𝑌 ≤ 𝑋 ↔ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
12		brxp 5712	. . . . . 6 ⊢ (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
13	4, 3, 12	sylanbrc 595	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
14		brin 5165	. . . . . 6 ⊢ (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋 ∧ 𝑌(𝐵 × 𝐵)𝑋))
15	14	rbaib 548	. . . . 5 ⊢ (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
16	13, 15	syl 18	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
17	11, 16	bitrid 286	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ≤ 𝑋 ↔ 𝑌(le‘𝐾)𝑋))
18	10, 17	anbi12d 644	. 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 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
21	19, 20	posasymb 18399	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌))
22	18, 21	bitrd 282	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∩ cin 3905   class class class wbr 5111   × cxp 5661  ‘cfv 6540  Basecbs 17293  lecple 17341  Posetcpo 18387

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-iota 6496  df-fv 6548  df-proset 18374  df-poset 18393

This theorem is used by:  ordtconnlem1  34380