Theorem elnanel 9650

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9649 and serves as an example in the context of Godel codes, see elnanelprv 35257. (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 9649	. 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		df-nan 1486	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpbir 230	1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 394   ⊼ wnan 1485   ∈ wcel 2099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-reg 9635

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-nan 1486  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-eprel 5586  df-fr 5637

This theorem is referenced by:  elnanelprv  35257