Theorem bj-nsnid 37065

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 4684): ⊢ ¬ ({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 9566	. 2 ⊢ ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)
2		snidg 4627	. . . 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 2109  {csn 4592

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-eprel 5541  df-fr 5594

This theorem is referenced by:  bj-inftyexpitaudisj  37200