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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 4040 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | 1 | con3i 600 | . . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V) |
3 | snexprc 4042 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V) |
5 | 4 | con3i 600 | . 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) |
6 | ax-in1 582 | . 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V) | |
7 | 5, 6 | ax-mp 7 | 1 ⊢ ¬ ¬ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1445 Vcvv 2633 {csn 3466 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 |
This theorem is referenced by: (None) |
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