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| Mirrors > Home > MPE Home > Th. List > 0tsk | Structured version Visualization version GIF version | ||
| Description: The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.) |
| Ref | Expression |
|---|---|
| 0tsk | ⊢ ∅ ∈ Tarski |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ral0 4513 | . 2 ⊢ ∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) | |
| 2 | elsni 4643 | . . . . 5 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
| 3 | 0ex 5307 | . . . . . . . 8 ⊢ ∅ ∈ V | |
| 4 | 3 | enref 9025 | . . . . . . 7 ⊢ ∅ ≈ ∅ |
| 5 | breq1 5146 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅)) | |
| 6 | 4, 5 | mpbiri 258 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ≈ ∅) |
| 7 | 6 | orcd 874 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
| 8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
| 9 | pw0 4812 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
| 10 | 8, 9 | eleq2s 2859 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
| 11 | 10 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅) |
| 12 | eltsk2g 10791 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ Tarski ↔ (∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) ∧ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)))) | |
| 13 | 3, 12 | ax-mp 5 | . 2 ⊢ (∅ ∈ Tarski ↔ (∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) ∧ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅))) |
| 14 | 1, 11, 13 | mpbir2an 711 | 1 ⊢ ∅ ∈ Tarski |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 ≈ cen 8982 Tarskictsk 10788 |
| 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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 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-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 df-tsk 10789 |
| This theorem is referenced by: r1tskina 10822 grutsk 10862 tskmcl 10881 |
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