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| Mirrors > Home > MPE Home > Th. List > eltskg | Structured version Visualization version GIF version | ||
| Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) |
| Ref | Expression |
|---|---|
| eltskg | ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq2 4010 | . . . . 5 ⊢ (𝑦 = 𝑇 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑧 ⊆ 𝑇)) | |
| 2 | rexeq 3322 | . . . . 5 ⊢ (𝑦 = 𝑇 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)) | |
| 3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝑦 = 𝑇 → ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))) |
| 4 | 3 | raleqbi1dv 3338 | . . 3 ⊢ (𝑦 = 𝑇 → (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))) |
| 5 | pweq 4614 | . . . 4 ⊢ (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇) | |
| 6 | breq2 5147 | . . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ≈ 𝑦 ↔ 𝑧 ≈ 𝑇)) | |
| 7 | eleq2 2830 | . . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑇)) | |
| 8 | 6, 7 | orbi12d 919 | . . . 4 ⊢ (𝑦 = 𝑇 → ((𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))) |
| 9 | 5, 8 | raleqbidv 3346 | . . 3 ⊢ (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))) |
| 10 | 4, 9 | anbi12d 632 | . 2 ⊢ (𝑦 = 𝑇 → ((∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))) |
| 11 | df-tsk 10789 | . 2 ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))} | |
| 12 | 10, 11 | elab2g 3680 | 1 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 𝒫 cpw 4600 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-ext 2708 |
| 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-sb 2065 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-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-tsk 10789 |
| This theorem is referenced by: eltsk2g 10791 tskpwss 10792 tsken 10794 grothtsk 10875 |
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