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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 4184 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 {csn 3592 |
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-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 4121 ax-pow 4174 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 |
This theorem is referenced by: snelpw 4213 rext 4215 sspwb 4216 intid 4224 euabex 4225 mss 4226 exss 4227 opi1 4232 opeqsn 4252 opeqpr 4253 uniop 4255 snnex 4448 op1stb 4478 dtruex 4558 relop 4777 funopg 5250 fo1st 6157 fo2nd 6158 mapsn 6689 mapsnconst 6693 mapsncnv 6694 mapsnf1o2 6695 elixpsn 6734 ixpsnf1o 6735 ensn1 6795 mapsnen 6810 xpsnen 6820 endisj 6823 xpcomco 6825 xpassen 6829 phplem2 6852 findcard2 6888 findcard2s 6889 ac6sfi 6897 xpfi 6928 djuex 7041 0ct 7105 finomni 7137 exmidfodomrlemim 7199 djuassen 7215 cc2lem 7264 nn0ex 9181 fxnn0nninf 10437 inftonninf 10440 hashxp 10805 reldvg 14118 |
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