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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 4765 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snprc 4717 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 215 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 1, 3 | sylan9eq 2788 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∅c0 4318 {csn 4624 {cpr 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-dif 3948 df-un 3950 df-nul 4319 df-sn 4625 df-pr 4627 |
This theorem is referenced by: (None) |
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