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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 4028 | . 2 ⊢ ∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) | |
2 | elsni 4142 | . . . . 5 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
3 | 0ex 4718 | . . . . . . . 8 ⊢ ∅ ∈ V | |
4 | 3 | enref 7874 | . . . . . . 7 ⊢ ∅ ≈ ∅ |
5 | breq1 4586 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅)) | |
6 | 4, 5 | mpbiri 247 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ≈ ∅) |
7 | 6 | orcd 406 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
9 | pw0 4283 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | 8, 9 | eleq2s 2706 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
11 | 10 | rgen 2906 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅) |
12 | eltsk2g 9452 | . . 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 957 | 1 ⊢ ∅ ∈ Tarski |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ≈ cen 7838 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-eu 2462 df-mo 2463 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-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-en 7842 df-tsk 9450 |
This theorem is referenced by: r1tskina 9483 grutsk 9523 tskmcl 9542 |
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