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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 4468 | . 2 ⊢ ∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) | |
2 | elsni 4601 | . . . . 5 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
3 | 0ex 5262 | . . . . . . . 8 ⊢ ∅ ∈ V | |
4 | 3 | enref 8921 | . . . . . . 7 ⊢ ∅ ≈ ∅ |
5 | breq1 5106 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅)) | |
6 | 4, 5 | mpbiri 257 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ≈ ∅) |
7 | 6 | orcd 871 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
9 | pw0 4770 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | 8, 9 | eleq2s 2856 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
11 | 10 | rgen 3064 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅) |
12 | eltsk2g 10683 | . . 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 709 | 1 ⊢ ∅ ∈ Tarski |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∀wral 3062 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 {csn 4584 class class class wbr 5103 ≈ cen 8876 Tarskictsk 10680 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-en 8880 df-tsk 10681 |
This theorem is referenced by: r1tskina 10714 grutsk 10754 tskmcl 10773 |
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