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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 4722): ⊢ ¬ ({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 9644 | . 2 ⊢ ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴) | |
2 | snidg 4665 | . . . 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 2106 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-fr 5641 |
This theorem is referenced by: bj-inftyexpitaudisj 37188 |
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