Theorem elnanel 9365

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9364 and serves as an example in the context of Godel codes, see elnanelprv 33391. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
elnanel	⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Proof of Theorem elnanel

Step	Hyp	Ref	Expression
1		en2lp 9364	. 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		df-nan 1487	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpbir 230	1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 396   ⊼ wnan 1486   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-nan 1487  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-fr 5544

This theorem is referenced by:  elnanelprv  33391