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Mirrors > Home > MPE Home > Th. List > tsksn | Structured version Visualization version GIF version |
Description: A singleton of an element of a Tarski class belongs to the class. JFM CLASSES2 th. 2 (partly). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.) |
Ref | Expression |
---|---|
tsksn | ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tskpw 10776 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | |
2 | snsspw 4846 | . . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴 | |
3 | tskss 10781 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇) | |
4 | 2, 3 | mp3an3 1447 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) |
5 | 1, 4 | syldan 590 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ⊆ wss 3947 𝒫 cpw 4603 {csn 4629 Tarskictsk 10771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-pow 5365 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-tsk 10772 |
This theorem is referenced by: tsk1 10787 tskop 10794 |
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