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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 3637 | . 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) | |
3 | 1, 2 | mpbi 145 | 1 ⊢ 𝐴 ∈ {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 Vcvv 2752 {csn 3607 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sn 3613 |
This theorem is referenced by: vsnid 3639 exsnrex 3649 rabsnt 3682 sneqr 3775 undifexmid 4211 exmidexmid 4214 ss1o0el1 4215 exmidundif 4224 exmidundifim 4225 exmid1stab 4226 unipw 4235 intid 4242 ordtriexmidlem2 4537 ordtriexmid 4538 ontriexmidim 4539 ordtri2orexmid 4540 regexmidlem1 4550 0elsucexmid 4582 ordpwsucexmid 4587 opthprc 4695 fsn 5708 fsn2 5710 fvsn 5731 fvsnun1 5733 acexmidlema 5886 acexmidlemb 5887 acexmidlemab 5889 brtpos0 6276 mapsn 6715 mapsncnv 6720 0elixp 6754 en1 6824 djulclr 7077 djurclr 7078 djulcl 7079 djurcl 7080 djuf1olem 7081 exmidonfinlem 7221 elreal2 7858 1exp 10579 hashinfuni 10788 ennnfonelemhom 12465 dvef 14640 djucllem 15005 bj-d0clsepcl 15130 |
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