Theorem n0i 3497

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2816. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
n0i	⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3495	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2293	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 679	. 2 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)
4	3	con2i 630	1 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ∈ wcel 2200  ∅c0 3491

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-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  ne0i  3498  n0ii  3500  unidif0  4250  iin0r  4252  nnm00  6674  dif1enen  7038  enq0tr  7617  gsum0g  13424  gsumval2  13425