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Mirrors > Home > MPE Home > Th. List > prprc | Structured version Visualization version 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 4762 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snprc 4714 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 215 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 1, 3 | sylan9eq 2784 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 {cpr 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3944 df-un 3946 df-nul 4316 df-sn 4622 df-pr 4624 |
This theorem is referenced by: (None) |
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