Theorem eldifsnbd 4748

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 4747	. . 3 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
4	3	simprd 501	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   ≠ wne 2955   ∖ cdif 4045  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-sn 4584

This theorem is used by:  isdrng2  20997  lgsquadlem1  27707  ragsupplcgra  29345  angmgmaddlid  29392  angmgmaddrid  29393  angmgmlem  29395  dfprlng2  29425  frlmnzcoordex  43653  frlmnzcoordsca  43659  prjspnnorm  43662  chnsubseq  47890