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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 3587 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | imbi1i 237 | . . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | 2 | albii 1457 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
4 | dfss2 3126 | . 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
5 | snss.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | clel2 2854 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 3, 4, 6 | 3bitr4ri 212 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 {csn 3570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 df-sn 3576 |
This theorem is referenced by: snssg 3703 prss 3723 tpss 3732 snelpw 4185 sspwb 4188 mss 4198 exss 4199 reg2exmidlema 4505 elomssom 4576 relsn 4703 fnressn 5665 un0mulcl 9139 nn0ssz 9200 fimaxre2 11154 fsum2dlemstep 11361 fsumabs 11392 fsumiun 11404 fprod2dlemstep 11549 bdsnss 13590 |
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