Theorem snnzg 3784

Description: The singleton of a set is not empty. It is also inhabited as shown at snmg 3785. (Contributed by NM, 14-Dec-2008.)

Assertion

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

Proof of Theorem snnzg

Step	Hyp	Ref	Expression
1		snidg 3695	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		ne0i 3498	. 2 ⊢ (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491  {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by:  snnz  3786  0nelop  4334