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Theorem bj-nsnid 35532
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 4678): ¬ ({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 9541 . 2 ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)
2 snidg 4620 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴})
32anim1i 615 . . 3 ((𝐴𝑉 ∧ {𝐴} ∈ 𝐴) → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴))
43ex 413 . 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 396  wcel 2106  {csn 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-reg 9527
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-eprel 5537  df-fr 5588
This theorem is referenced by:  bj-inftyexpitaudisj  35667
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