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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 2097 | . 2 ⊢ (x = A → (x ∈ B ↔ A ∈ B)) | |
2 | sneq 3378 | . . 3 ⊢ (x = A → {x} = {A}) | |
3 | 2 | sseq1d 2966 | . 2 ⊢ (x = A → ({x} ⊆ B ↔ {A} ⊆ B)) |
4 | vex 2554 | . . 3 ⊢ x ∈ V | |
5 | 4 | snss 3485 | . 2 ⊢ (x ∈ B ↔ {x} ⊆ B) |
6 | 1, 3, 5 | vtoclbg 2608 | 1 ⊢ (A ∈ 𝑉 → (A ∈ B ↔ {A} ⊆ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-sn 3373 |
This theorem is referenced by: snssi 3499 snssd 3500 prssg 3512 ordtri2orexmid 4211 fvimacnvi 5224 fvimacnv 5225 |
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