Theorem elnanel 9067

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9066 and serves as an example in the context of Godel codes, see elnanelprv 32700. (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 9066	. 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		df-nan 1481	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpbir 233	1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 398   ⊼ wnan 1480   ∈ wcel 2113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pr 5327  ax-reg 9053

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-nan 1481  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-eprel 5462  df-fr 5511

This theorem is referenced by:  elnanelprv  32700