Theorem eldifsnbd 4749

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 4748	. . 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 2147   ≠ wne 2959   ∖ cdif 4048  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-sn 4589

This theorem is used by:  isdrng2  21024  lgsquadlem1  27734  ragsupplcgra  29372  angmgmaddlid  29419  angmgmaddrid  29420  angmgmlem  29422  dfprlng2  29452  frlmnzcoordex  43843  frlmnzcoordsca  43849  prjspnnorm  43852  chnsubseq  48093