Theorem prneli 4621

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.)

Hypotheses

Ref	Expression
prneli.1	⊢ 𝐴 ≠ 𝐵
prneli.2	⊢ 𝐴 ≠ 𝐶

Assertion

Ref	Expression
prneli	⊢ 𝐴 ∉ {𝐵, 𝐶}

Proof of Theorem prneli

Step	Hyp	Ref	Expression
1		prneli.1	. . 3 ⊢ 𝐴 ≠ 𝐵
2		prneli.2	. . 3 ⊢ 𝐴 ≠ 𝐶
3	1, 2	nelpri 4620	. 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
4	3	nelir 3065	1 ⊢ 𝐴 ∉ {𝐵, 𝐶}

Syntax hints:   ≠ wne 2956   ∉ wnel 3062  {cpr 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-v 3455  df-un 3909  df-sn 4589  df-pr 4591

This theorem is referenced by:  vdegp1ai  29852