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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 4196 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | 1 | con3i 633 | . . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V) |
3 | snexprc 4198 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V) |
5 | 4 | con3i 633 | . 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) |
6 | pm2.01 617 | . 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 2158 Vcvv 2749 {csn 3604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-dif 3143 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 |
This theorem is referenced by: (None) |
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