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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 4613): ⊢ ¬ ({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 9053 | . 2 ⊢ ¬ (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴) | |
2 | snidg 4559 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
3 | 2 | anim1i 617 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ 𝐴) → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴)) |
4 | 3 | ex 416 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ 𝐴 → (𝐴 ∈ {𝐴} ∧ {𝐴} ∈ 𝐴))) |
5 | 1, 4 | mtoi 202 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 |
This theorem is referenced by: bj-inftyexpitaudisj 34620 |
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