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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 4770 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snprc 4722 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 216 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 1, 3 | sylan9eq 2795 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-pr 4634 |
This theorem is referenced by: (None) |
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