Theorem eldifsnd 4754

Description: Membership in a set with an element removed : deduction version. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
eldifsnd.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
eldifsnd.2	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Assertion

Ref	Expression
eldifsnd	⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))

Proof of Theorem eldifsnd

Step	Hyp	Ref	Expression
1		eldifsnd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		eldifsnd.2	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
3		eldifsn 4753	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2926   ∖ cdif 3914  {csn 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-sn 4593

This theorem is referenced by:  prproe  4872  drngmcl  20666  r1pid2  26074  pfxchn  32942  chnind  32944  irrednzr  33208  fracfld  33265  rprmasso2  33504  rprmirredlem  33508  1arithidomlem1  33513  ufdprmidl  33519  1arithufdlem3  33524  1arithufdlem4  33525  dfufd2lem  33527  dfufd2  33528  zringfrac  33532  ply1dg1rt  33555  r1peuqusdeg1  35637  unitscyglem4  42193  resuppsinopn  42358  readvcot  42359  redivvald  42437  domnexpgn0cl  42518  drngmullcan  42520  drngmulrcan  42521  prjspvs  42605