Theorem difeqri 3349

Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypothesis

Ref	Expression
difeqri.1	⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)

Assertion

Ref	Expression
difeqri	⊢ (𝐴 ∖ 𝐵) = 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem difeqri

Step	Hyp	Ref	Expression
1		eldif 3229	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		difeqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 184	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2235	1 ⊢ (𝐴 ∖ 𝐵) = 𝐶

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ∖ cdif 3217

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222

This theorem is referenced by:  difdif  3354  ddifnel  3360  difab  3500