brdifun - Intuitionistic Logic Explorer

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Theorem brdifun 6616

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 4692	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
2		df-br 4031	. . . 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 4036	. . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ^◡ < ))𝐵)
6		brdif 4083	. . . . 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 920	. . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
9	3, 8	syl 14	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ^◡ < )𝐵))
10		brun 4081	. . . 4 ⊢ (𝐴( < ∪ ^◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴^◡ < 𝐵))
11		brcnvg 4844	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴^◡ < 𝐵 ↔ 𝐵 < 𝐴))
12	11	orbi2d 791	. . . 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 668	. 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 709 = wceq 1364 ∈ wcel 2164 ∖ cdif 3151 ∪ cun 3152 ⟨cop 3622 class class class wbr 4030 × cxp 4658 ^◡ccnv 4659

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-br 4031 df-opab 4092 df-xp 4666 df-cnv 4668

This theorem is referenced by: swoer 6617 swoord1 6618 swoord2 6619

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