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Theorem elnanel 9551
Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9550 and serves as an example in the context of Godel codes, see elnanelprv 34087. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)
Assertion
Ref Expression
elnanel (𝐴𝐵𝐵𝐴)

Proof of Theorem elnanel
StepHypRef Expression
1 en2lp 9550 . 2 ¬ (𝐴𝐵𝐵𝐴)
2 df-nan 1491 . 2 ((𝐴𝐵𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
31, 2mpbir 230 1 (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wnan 1490  wcel 2107
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-nan 1491  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-eprel 5541  df-fr 5592
This theorem is referenced by:  elnanelprv  34087
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