Theorem snnzg 3789

Description: The singleton of a set is not empty. It is also inhabited as shown at snmg 3790. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
snnzg	⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Proof of Theorem snnzg

Step	Hyp	Ref	Expression
1		snidg 3698	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		ne0i 3501	. 2 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2202   ≠ wne 2402  ∅c0 3494  {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-nul 3495  df-sn 3675

This theorem is referenced by:  snnz  3791  0nelop  4340