Theorem elnanel 9497

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9496 and serves as an example in the context of Godel codes, see elnanelprv 35473. (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 9496	. 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		df-nan 1493	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpbir 231	1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   ⊼ wnan 1492   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-nan 1493  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567

This theorem is referenced by:  elnanelprv  35473