Theorem snssg 3491

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.)

Assertion

Ref	Expression
snssg	⊢ (A ∈ 𝑉 → (A ∈ B ↔ {A} ⊆ B))

Proof of Theorem snssg

Dummy variable x is distinct from all other variables.

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))

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-bnd 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