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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 4703 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snprc 4655 | . . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 218 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 1, 3 | sylan9eq 2878 | 1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {csn 4569 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: (None) |
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