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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 4459 | . 2 ⊢ ∀𝑥 ∈ ∅ (𝒫 𝑥 ⊆ ∅ ∧ 𝒫 𝑥 ∈ ∅) | |
2 | elsni 4587 | . . . . 5 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
3 | 0ex 5214 | . . . . . . . 8 ⊢ ∅ ∈ V | |
4 | 3 | enref 8545 | . . . . . . 7 ⊢ ∅ ≈ ∅ |
5 | breq1 5072 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅)) | |
6 | 4, 5 | mpbiri 260 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ≈ ∅) |
7 | 6 | orcd 869 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
9 | pw0 4748 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | 8, 9 | eleq2s 2934 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∅ → (𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅)) |
11 | 10 | rgen 3151 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ∅(𝑥 ≈ ∅ ∨ 𝑥 ∈ ∅) |
12 | eltsk2g 10176 | . . 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 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ⊆ wss 3939 ∅c0 4294 𝒫 cpw 4542 {csn 4570 class class class wbr 5069 ≈ cen 8509 Tarskictsk 10173 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-en 8513 df-tsk 10174 |
This theorem is referenced by: r1tskina 10207 grutsk 10247 tskmcl 10266 |
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