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Theorem bj-nsnid 35951
Description: A set does not contain the singleton formed on it. More precisely, one can prove that a class contains the singleton formed on it if and only if it is proper and contains the empty set (since it is "the singleton formed on" any proper class, see snprc 4722): ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
Assertion
Ref Expression
bj-nsnid (𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)

Proof of Theorem bj-nsnid
StepHypRef Expression
1 en2lp 9601 . 2 ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)
2 snidg 4663 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴})
32anim1i 616 . . 3 ((𝐴𝑉 ∧ {𝐴} ∈ 𝐴) → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴))
43ex 414 . 2 (𝐴𝑉 → ({𝐴} ∈ 𝐴 → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)))
51, 4mtoi 198 1 (𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  {csn 4629
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 5300  ax-nul 5307  ax-pr 5428  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632
This theorem is referenced by:  bj-inftyexpitaudisj  36086
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