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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 2684 | . 2 ⊢ 𝑥 ∈ V | |
2 | 1 | elsn 3538 | 1 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 {csn 3522 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sn 3528 |
This theorem is referenced by: dfpr2 3541 mosn 3555 ralsnsg 3556 ralsns 3557 rexsns 3558 disjsn 3580 snprc 3583 euabsn2 3587 prmg 3639 snss 3644 difprsnss 3653 eqsnm 3677 snsssn 3683 snsspw 3686 dfnfc2 3749 uni0b 3756 uni0c 3757 sndisj 3920 unidif0 4086 exmid01 4116 rext 4132 exss 4144 frirrg 4267 ordsucim 4411 ordtriexmidlem 4430 ordtri2or2exmidlem 4436 onsucelsucexmidlem 4439 elirr 4451 sucprcreg 4459 fconstmpt 4581 opeliunxp 4589 dmsnopg 5005 dfmpt3 5240 nfunsn 5448 fsn 5585 fnasrn 5591 fnasrng 5593 fconstfvm 5631 eusvobj2 5753 opabex3d 6012 opabex3 6013 dcdifsnid 6393 ecexr 6427 ixp0x 6613 xpsnen 6708 fidifsnen 6757 difinfsn 6978 exmidonfinlem 7042 iccid 9701 fzsn 9839 fzpr 9850 fzdifsuc 9854 fsum2dlemstep 11196 ef0lem 11355 1nprm 11784 restsn 12338 |
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