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Mirrors > Home > ILE Home > Th. List > notnotsnex | GIF version |
Description: A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.) |
Ref | Expression |
---|---|
notnotsnex | ⊢ ¬ ¬ {𝐴} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snexg 4157 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | 1 | con3i 622 | . . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V) |
3 | snexprc 4159 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V) |
5 | 4 | con3i 622 | . 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) |
6 | pm2.01 606 | . 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ ¬ ¬ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2135 Vcvv 2721 {csn 3570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 |
This theorem is referenced by: (None) |
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