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Theorem brdifun 6040

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 4299	. . . 4 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → ⟨A, B⟩ ∈ (𝑋 × 𝑋))
2		df-br 3735	. . . 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 3740	. . . . 5 ⊢ (A𝑅B ↔ A((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))B)
6		brdif 3782	. . . . 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 816	. . 3 ⊢ (A(𝑋 × 𝑋)B → (A𝑅B ↔ ¬ A( < ∪ ^◡ < )B))
9	3, 8	syl 14	. 2 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A𝑅B ↔ ¬ A( < ∪ ^◡ < )B))
10		brun 3780	. . . 4 ⊢ (A( < ∪ ^◡ < )B ↔ (A < B ∨ A^◡ < B))
11		brcnvg 4439	. . . . 5 ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A^◡ < B ↔ B < A))
12	11	orbi2d 691	. . . 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 579	. 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 616 = wceq 1226 ∈ wcel 1370 ∖ cdif 2887 ∪ cun 2888 ⟨cop 3349 class class class wbr 3734 × cxp 4266 ^◡ccnv 4267

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 532 ax-in2 533 ax-io 617 ax-5 1312 ax-7 1313 ax-gen 1314 ax-ie1 1359 ax-ie2 1360 ax-8 1372 ax-10 1373 ax-11 1374 ax-i12 1375 ax-bnd 1376 ax-4 1377 ax-14 1382 ax-17 1396 ax-i9 1400 ax-ial 1405 ax-i5r 1406 ax-ext 2000 ax-sep 3845 ax-pow 3897 ax-pr 3914

This theorem depends on definitions: df-bi 110 df-3an 873 df-tru 1229 df-nf 1326 df-sb 1624 df-eu 1881 df-mo 1882 df-clab 2005 df-cleq 2011 df-clel 2014 df-nfc 2145 df-ral 2285 df-rex 2286 df-v 2533 df-dif 2893 df-un 2895 df-in 2897 df-ss 2904 df-pw 3332 df-sn 3352 df-pr 3353 df-op 3355 df-br 3735 df-opab 3789 df-xp 4274 df-cnv 4276

This theorem is referenced by: swoer 6041 swoord1 6042 swoord2 6043

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