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Mirrors > Home > ILE Home > Th. List > brdifun | GIF version |
Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
swoer.1 | ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) |
Ref | Expression |
---|---|
brdifun | ⊢ ((A ∈ 𝑋 ∧ B ∈ 𝑋) → (A𝑅B ↔ ¬ (A < B ∨ B < A))) |
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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