Theorem elnanel 9572

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9571 and serves as an example in the context of Godel codes, see elnanelprv 35921. (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 9571	. 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		df-nan 1522	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpbir 234	1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 400   ⊼ wnan 1521   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-nan 1522  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614

This theorem is referenced by:  elnanelprv  35921