brdifun - Intuitionistic Logic Explorer

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Theorem brdifun 6069

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 4319	. . . 4 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → ⟨A, B⟩ ∈ (𝑋 × 𝑋))
2		df-br 3756	. . . 4 ⊢ (A(𝑋 × 𝑋)B ↔ ⟨A, B⟩ ∈ (𝑋 × 𝑋))
3	1, 2	sylibr 137	. . 3 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → A(𝑋 × 𝑋)B)
4		swoer.1	. . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))
5	4	breqi 3761	. . . . 5 ⊢ (A𝑅B ↔ A((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))B)
6		brdif 3803	. . . . 5 ⊢ (A((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))B ↔ (A(𝑋 × 𝑋)B ∧ ¬ A( < ∪ ^◡ < )B))
7	5, 6	bitri 173	. . . 4 ⊢ (A𝑅B ↔ (A(𝑋 × 𝑋)B ∧ ¬ A( < ∪ ^◡ < )B))
8	7	baib 827	. . 3 ⊢ (A(𝑋 × 𝑋)B → (A𝑅B ↔ ¬ A( < ∪ ^◡ < )B))
9	3, 8	syl 14	. 2 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A𝑅B ↔ ¬ A( < ∪ ^◡ < )B))
10		brun 3801	. . . 4 ⊢ (A( < ∪ ^◡ < )B ↔ (A < B ∨ A^◡ < B))
11		brcnvg 4459	. . . . 5 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A^◡ < B ↔ B < A))
12	11	orbi2d 703	. . . 4 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → ((A < B ∨ A^◡ < B) ↔ (A < B ∨ B < A)))
13	10, 12	syl5bb 181	. . 3 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A( < ∪ ^◡ < )B ↔ (A < B ∨ B < A)))
14	13	notbid 591	. 2 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (¬ A( < ∪ ^◡ < )B ↔ ¬ (A < B ∨ B < A)))
15	9, 14	bitrd 177	1 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A𝑅B ↔ ¬ (A < B ∨ B < A)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∖ cdif 2908 ∪ cun 2909 ⟨cop 3370 class class class wbr 3755 × cxp 4286 ^◡ccnv 4287

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296

This theorem is referenced by: swoer 6070 swoord1 6071 swoord2 6072

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