Theorem n0i 3470

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2790. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3468	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2270	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 677	. 2 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)
4	3	con2i 628	1 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1373   ∈ wcel 2177  ∅c0 3464

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-nul 3465

This theorem is referenced by:  ne0i  3471  n0ii  3473  unidif0  4219  iin0r  4221  nnm00  6629  dif1enen  6992  enq0tr  7567  gsum0g  13303  gsumval2  13304