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Mirrors > Home > ILE Home > Th. List > snid | GIF version |
Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
snid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snid | ⊢ 𝐴 ∈ {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snid.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | snidb 3474 | . 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) | |
3 | 1, 2 | mpbi 143 | 1 ⊢ 𝐴 ∈ {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 Vcvv 2619 {csn 3446 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-sn 3452 |
This theorem is referenced by: vsnid 3476 exsnrex 3485 rabsnt 3517 sneqr 3604 undifexmid 4028 exmidexmid 4031 exmid01 4032 exmidundif 4035 unipw 4044 intid 4051 ordtriexmidlem2 4337 ordtriexmid 4338 ordtri2orexmid 4339 regexmidlem1 4349 0elsucexmid 4381 ordpwsucexmid 4386 opthprc 4489 fsn 5469 fsn2 5471 fvsn 5492 fvsnun1 5494 acexmidlema 5643 acexmidlemb 5644 acexmidlemab 5646 brtpos0 6017 mapsn 6447 mapsncnv 6452 en1 6516 djulclr 6741 djurclr 6742 djulcl 6743 djurcl 6744 djuf1olem 6745 elreal2 7368 1exp 9984 hashinfuni 10185 djucllem 11700 bj-d0clsepcl 11820 |
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