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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 4196 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2158 Vcvv 2749 {csn 3604 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 |
This theorem is referenced by: snelpw 4225 rext 4227 sspwb 4228 intid 4236 euabex 4237 mss 4238 exss 4239 opi1 4244 opeqsn 4264 opeqpr 4265 uniop 4267 snnex 4460 op1stb 4490 dtruex 4570 relop 4789 funopg 5262 fo1st 6171 fo2nd 6172 mapsn 6703 mapsnconst 6707 mapsncnv 6708 mapsnf1o2 6709 elixpsn 6748 ixpsnf1o 6749 ensn1 6809 mapsnen 6824 xpsnen 6834 endisj 6837 xpcomco 6839 xpassen 6843 phplem2 6866 findcard2 6902 findcard2s 6903 ac6sfi 6911 xpfi 6942 djuex 7055 0ct 7119 finomni 7151 exmidfodomrlemim 7213 djuassen 7229 cc2lem 7278 nn0ex 9195 fxnn0nninf 10451 inftonninf 10454 hashxp 10819 reldvg 14419 |
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