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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 4181 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | 1 | con3i 632 | . . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V) |
3 | snexprc 4183 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V) |
5 | 4 | con3i 632 | . 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) |
6 | pm2.01 616 | . 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 2148 Vcvv 2737 {csn 3591 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 |
This theorem is referenced by: (None) |
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