Theorem bj-nsnid 37125

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 4671): ⊢ ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴 → 𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)

Assertion

Ref	Expression
bj-nsnid	⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴)

Proof of Theorem bj-nsnid

Step	Hyp	Ref	Expression
1		en2lp 9506	. 2 ⊢ ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)
2		snidg 4614	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	2	anim1i 615	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ 𝐴) → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴))
4	3	ex 412	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ 𝐴 → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)))
5	1, 4	mtoi 199	1 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2113  {csn 4577

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-reg 9488

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-eprel 5521  df-fr 5574

This theorem is referenced by:  bj-inftyexpitaudisj  37260