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Mirrors > Home > ILE Home > Th. List > opprc | GIF version |
Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opprc | ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-op 3531 | . 2 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
2 | 3simpa 978 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 2 | con3i 621 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) |
4 | 3 | alrimiv 1846 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑥 ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) |
5 | abeq0 3388 | . . 3 ⊢ ({𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅ ↔ ∀𝑥 ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
6 | 4, 5 | sylibr 133 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅) |
7 | 1, 6 | syl5eq 2182 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 962 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2123 Vcvv 2681 ∅c0 3358 {csn 3522 {cpr 3523 〈cop 3525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-nul 3359 df-op 3531 |
This theorem is referenced by: opprc1 3722 opprc2 3723 ovprc 5799 |
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