|   | Mathbox for BJ | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nsnid | Structured version Visualization version GIF version | ||
| 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 4717): ⊢ ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴 → 𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.) | 
| Ref | Expression | 
|---|---|
| bj-nsnid | ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | en2lp 9646 | . 2 ⊢ ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴) | |
| 2 | snidg 4660 | . . . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 {csn 4626 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 | 
| This theorem is referenced by: bj-inftyexpitaudisj 37206 | 
| Copyright terms: Public domain | W3C validator |