Theorem prneli 4678

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 4677	. 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
4	3	nelir 3055	1 ⊢ 𝐴 ∉ {𝐵, 𝐶}

Syntax hints:   ≠ wne 2946   ∉ wnel 3052  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-nel 3053  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by:  vdegp1ai  29572