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Mirrors > Home > MPE Home > Th. List > Mathboxes > vsn | Structured version Visualization version GIF version |
Description: The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
vsn | ⊢ {V} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 5193 | . 2 ⊢ ¬ V ∈ V | |
2 | snprc 4618 | . 2 ⊢ (¬ V ∈ V ↔ {V} = ∅) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ {V} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2114 Vcvv 3400 ∅c0 4221 {csn 4526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-dif 3856 df-nul 4222 df-sn 4527 |
This theorem is referenced by: mo0 45738 |
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