Theorem prneli 4661

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 4660	. 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
4	3	nelir 3047	1 ⊢ 𝐴 ∉ {𝐵, 𝐶}

Syntax hints:   ≠ wne 2938   ∉ wnel 3044  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by:  vdegp1ai  29569