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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 3977 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1434 Vcvv 2611 {csn 3417 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2613 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 |
This theorem is referenced by: snelpw 3998 rext 4000 sspwb 4001 intid 4009 euabex 4010 mss 4011 exss 4012 opi1 4017 opeqsn 4037 opeqpr 4038 uniop 4040 snnex 4229 op1stb 4257 dtruex 4332 relop 4537 funopg 4987 fo1st 5841 fo2nd 5842 ensn1 6371 xpsnen 6393 endisj 6396 xpcomco 6398 xpassen 6402 phplem2 6416 findcard2 6452 findcard2s 6453 ac6sfi 6461 xpfi 6479 djuex 6560 finomni 6590 nn0ex 8447 hashxp 9936 |
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