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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 4025 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ {𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 Vcvv 2620 {csn 3450 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 |
This theorem is referenced by: snelpw 4049 rext 4051 sspwb 4052 intid 4060 euabex 4061 mss 4062 exss 4063 opi1 4068 opeqsn 4088 opeqpr 4089 uniop 4091 snnex 4283 op1stb 4313 dtruex 4388 relop 4599 funopg 5061 fo1st 5942 fo2nd 5943 mapsn 6461 mapsnconst 6465 mapsncnv 6466 mapsnf1o2 6467 elixpsn 6506 ixpsnf1o 6507 ensn1 6567 mapsnen 6582 xpsnen 6591 endisj 6594 xpcomco 6596 xpassen 6600 phplem2 6623 findcard2 6659 findcard2s 6660 ac6sfi 6668 xpfi 6694 djuex 6790 0ct 6843 finomni 6857 exmidfodomrlemim 6888 nn0ex 8740 fxnn0nninf 9905 inftonninf 9908 hashxp 10295 |
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