Theorem eldifsnbd 4753

Description: Membership in a set with an element removed implies non-equality with that element. (Contributed by Thierry Arnoux, 13-Jul-2026.)

Hypothesis

Ref	Expression
eldifsnbd.1	⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))

Assertion

Ref	Expression
eldifsnbd	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem eldifsnbd

Step	Hyp	Ref	Expression
1		eldifsnbd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))
2		eldifsn 4752	. . 3 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
4	3	simprd 500	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142   ≠ wne 2957   ∖ cdif 3901  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-sn 4589

This theorem is used by:  ragsupplcgra  29159  dfprlng2  29208