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Mirrors > Home > ILE Home > Th. List > snss | 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, 5-Aug-1993.) |
Ref | Expression |
---|---|
snss.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snss | ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 3415 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | imbi1i 236 | . . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | 2 | albii 1399 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
4 | dfss2 2988 | . 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
5 | snss.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | clel2 2728 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 4, 6 | 3bitr4ri 211 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 {csn 3398 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-sn 3404 |
This theorem is referenced by: snssg 3522 prss 3541 tpss 3550 snelpw 3968 sspwb 3971 mss 3981 exss 3982 reg2exmidlema 4277 elnn 4346 relsn 4461 fnressn 5370 un0mulcl 8322 nn0ssz 8369 fimaxre2 10109 bdsnss 10664 |
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