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Mirrors > Home > ILE Home > Th. List > snex | GIF version |
Description: A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
snex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snex | ⊢ {𝐴} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | snexg 3936 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 Vcvv 2557 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 |
This theorem is referenced by: ensn1 6276 xpsnen 6295 endisj 6298 xpcomco 6300 xpassen 6304 phplem2 6316 findcard2 6346 findcard2s 6347 ac6sfi 6352 nn0ex 8187 |
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