Theorem snssg 3470

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 2078	. 2 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
2		sneq 3357	. . 3 ⊢ (x = A → {x} = {A})
3	2	sseq1d 2945	. 2 ⊢ (x = A → ({x} ⊆ B ↔ {A} ⊆ B))
4		vex 2534	. . 3 ⊢ x ∈ V
5	4	snss 3464	. 2 ⊢ (x ∈ B ↔ {x} ⊆ B)
6	1, 3, 5	vtoclbg 2587	1 ⊢ (A ∈ 𝑉 → (A ∈ B ↔ {A} ⊆ B))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370   ⊆ wss 2890  {csn 3346

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904  df-sn 3352

This theorem is referenced by:  snssi  3478  snssd  3479  prssg  3491  ordtri2orexmid  4191  fvimacnvi  5202  fvimacnv  5203