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Theorem bj-nsnid 34376
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 4627): ¬ ({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 9045 . 2 ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)
2 snidg 4573 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴})
32anim1i 616 . . 3 ((𝐴𝑉 ∧ {𝐴} ∈ 𝐴) → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴))
43ex 415 . 2 (𝐴𝑉 → ({𝐴} ∈ 𝐴 → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)))
51, 4mtoi 201 1 (𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wcel 2114  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304  ax-reg 9032
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-br 5041  df-opab 5103  df-eprel 5439  df-fr 5488
This theorem is referenced by:  bj-inftyexpitaudisj  34501
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