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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 9454 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | |
2 | snsspw 4315 | . . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴 | |
3 | tskss 9459 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇) | |
4 | 2, 3 | mp3an3 1405 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) |
5 | 1, 4 | syldan 486 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 Tarskictsk 9449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-tsk 9450 |
This theorem is referenced by: tsk1 9465 tskop 9472 |
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