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Mirrors > Home > ILE Home > Th. List > velsn | GIF version |
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
velsn | ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . 2 ⊢ 𝑥 ∈ V | |
2 | 1 | elsn 3543 | 1 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 {csn 3527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sn 3533 |
This theorem is referenced by: dfpr2 3546 mosn 3560 ralsnsg 3561 ralsns 3562 rexsns 3563 disjsn 3585 snprc 3588 euabsn2 3592 prmg 3644 snss 3649 difprsnss 3658 eqsnm 3682 snsssn 3688 snsspw 3691 dfnfc2 3754 uni0b 3761 uni0c 3762 sndisj 3925 unidif0 4091 exmid01 4121 rext 4137 exss 4149 frirrg 4272 ordsucim 4416 ordtriexmidlem 4435 ordtri2or2exmidlem 4441 onsucelsucexmidlem 4444 elirr 4456 sucprcreg 4464 fconstmpt 4586 opeliunxp 4594 dmsnopg 5010 dfmpt3 5245 nfunsn 5455 fsn 5592 fnasrn 5598 fnasrng 5600 fconstfvm 5638 eusvobj2 5760 opabex3d 6019 opabex3 6020 dcdifsnid 6400 ecexr 6434 ixp0x 6620 xpsnen 6715 fidifsnen 6764 difinfsn 6985 exmidonfinlem 7049 iccid 9708 fzsn 9846 fzpr 9857 fzdifsuc 9861 fsum2dlemstep 11203 ef0lem 11366 1nprm 11795 restsn 12349 |
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