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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 4410 | . 2 ⊢ ∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) | |
2 | elsni 4544 | . . . . 5 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
3 | 0ex 5185 | . . . . . . . 8 ⊢ ∅ ∈ V | |
4 | 3 | enref 8639 | . . . . . . 7 ⊢ ∅ ≈ ∅ |
5 | breq1 5042 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅)) | |
6 | 4, 5 | mpbiri 261 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ≈ ∅) |
7 | 6 | orcd 873 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
9 | pw0 4711 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | 8, 9 | eleq2s 2849 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
11 | 10 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅) |
12 | eltsk2g 10330 | . . 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 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 {csn 4527 class class class wbr 5039 ≈ cen 8601 Tarskictsk 10327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-en 8605 df-tsk 10328 |
This theorem is referenced by: r1tskina 10361 grutsk 10401 tskmcl 10420 |
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