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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 4199 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | 1 | con3i 633 | . . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V) |
3 | snexprc 4201 | . . . 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 2160 Vcvv 2752 {csn 3607 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 |
This theorem is referenced by: (None) |
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