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Mirrors > Home > ILE Home > Th. List > opprc2 | GIF version |
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3570. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opprc2 | ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | 1 | con3i 562 | . 2 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | opprc 3570 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
4 | 2, 3 | syl 14 | 1 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 〈cop 3378 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-nul 3225 df-op 3384 |
This theorem is referenced by: (None) |
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