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Theorem brdifun 8773

Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

**Hypothesis**
Ref	Expression
swoer.1	⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))

**Assertion**
Ref	Expression
brdifun	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

**Proof of Theorem brdifun**
Step	Hyp	Ref	Expression
1		opelxpi 5692	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
2		df-br 5107	. . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
3	1, 2	sylibr 237	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴(𝑋 × 𝑋)𝐵)
4		swoer.1	. . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))
5	4	breqi 5112	. . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))𝐵)
6		brdif 5161	. . . . 5 ⊢ (𝐴((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ^◡ < )𝐵))
7	5, 6	bitri 278	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ^◡ < )𝐵))
8	7	baib 545	. . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
9	3, 8	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
10		brun 5159	. . . 4 ⊢ (𝐴( < ∪ ^◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴^◡ < 𝐵))
11		brcnvg 5863	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴^◡ < 𝐵 ↔ 𝐵 < 𝐴))
12	11	orbi2d 929	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 < 𝐵 ∨ 𝐴^◡ < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
13	10, 12	bitrid 286	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( < ∪ ^◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
14	13	notbid 321	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (¬ 𝐴( < ∪ ^◡ < )𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
15	9, 14	bitrd 282	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ∖ cdif 4048 ∪ cun 4152 ⟨cop 4838 class class class wbr 5106 × cxp 5653 ^◡ccnv 5654

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5252 ax-pr 5395

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 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-br 5107 df-opab 5171 df-xp 5661 df-cnv 5663

This theorem is used by: swoer 8774 swoord1 8775 swoord2 8776 swoso 8777

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