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| 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 4765 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
| 2 | snprc 4717 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
| 3 | 2 | biimpi 216 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) | 
| 4 | 1, 3 | sylan9eq 2797 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: (None) | 
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