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Mirrors > Home > ILE Home > Th. List > prprc | GIF version |
Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
prprc | ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prprc1 3500 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snprc 3457 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 118 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 1, 3 | sylan9eq 2133 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∅c0 3251 {csn 3398 {cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-pr 3405 |
This theorem is referenced by: (None) |
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