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Theorem elnanel 9647
Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9646 and serves as an example in the context of Godel codes, see elnanelprv 35434. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)
Assertion
Ref Expression
elnanel (𝐴𝐵𝐵𝐴)

Proof of Theorem elnanel
StepHypRef Expression
1 en2lp 9646 . 2 ¬ (𝐴𝐵𝐵𝐴)
2 df-nan 1492 . 2 ((𝐴𝐵𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
31, 2mpbir 231 1 (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wnan 1491  wcel 2108
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-nan 1492  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637
This theorem is referenced by:  elnanelprv  35434
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