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Theorem brdifun 6558

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 4657	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
2		df-br 4003	. . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
3	1, 2	sylibr 134	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴(𝑋 × 𝑋)𝐵)
4		swoer.1	. . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))
5	4	breqi 4008	. . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))𝐵)
6		brdif 4055	. . . . 5 ⊢ (𝐴((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ^◡ < )𝐵))
7	5, 6	bitri 184	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ^◡ < )𝐵))
8	7	baib 919	. . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
9	3, 8	syl 14	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
10		brun 4053	. . . 4 ⊢ (𝐴( < ∪ ^◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴^◡ < 𝐵))
11		brcnvg 4806	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴^◡ < 𝐵 ↔ 𝐵 < 𝐴))
12	11	orbi2d 790	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 < 𝐵 ∨ 𝐴^◡ < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
13	10, 12	bitrid 192	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( < ∪ ^◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
14	13	notbid 667	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (¬ 𝐴( < ∪ ^◡ < )𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
15	9, 14	bitrd 188	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∖ cdif 3126 ∪ cun 3127 ⟨cop 3595 class class class wbr 4002 × cxp 4623 ^◡ccnv 4624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633

This theorem is referenced by: swoer 6559 swoord1 6560 swoord2 6561

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