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Mirrors > Home > ILE Home > Th. List > snssg | GIF version |
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) |
Ref | Expression |
---|---|
snssg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2151 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
2 | sneq 3463 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 2 | sseq1d 3056 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
4 | vex 2625 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | snss 3574 | . 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵) |
6 | 1, 3, 5 | vtoclbg 2683 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1290 ∈ wcel 1439 ⊆ wss 3002 {csn 3452 |
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-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
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 2624 df-in 3008 df-ss 3015 df-sn 3458 |
This theorem is referenced by: snssi 3589 snssd 3590 prssg 3602 ordtri2orexmid 4354 ordtri2or2exmid 4402 relsng 4556 fvimacnvi 5429 fvimacnv 5430 strslfv 11601 isneip 11909 elnei 11915 |
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