Theorem in0 3459

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
in0	⊢ (𝐴 ∩ ∅) = ∅

Proof of Theorem in0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3428	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
2	1	bianfi 947	. . 3 ⊢ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅))
3	2	bicomi 132	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅)
4	3	ineqri 3330	1 ⊢ (𝐴 ∩ ∅) = ∅

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ∩ cin 3130  ∅c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137  df-nul 3425

This theorem is referenced by:  0in  3460  res0  4913  dju0en  7215  rest0  13764