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| Mirrors > Home > MPE Home > Th. List > tskwe2 | Structured version Visualization version GIF version | ||
| Description: A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.) |
| Ref | Expression |
|---|---|
| tskwe2 | ⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elpwi 4607 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑇 → 𝑦 ⊆ 𝑇) | |
| 2 | tskssel 10797 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑦 ⊆ 𝑇 ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) | |
| 3 | 2 | 3exp 1120 | . . . . 5 ⊢ (𝑇 ∈ Tarski → (𝑦 ⊆ 𝑇 → (𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇))) |
| 4 | 1, 3 | syl5 34 | . . . 4 ⊢ (𝑇 ∈ Tarski → (𝑦 ∈ 𝒫 𝑇 → (𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇))) |
| 5 | 4 | ralrimiv 3145 | . . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑦 ∈ 𝒫 𝑇(𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇)) |
| 6 | rabss 4072 | . . 3 ⊢ ({𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝒫 𝑇(𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇)) | |
| 7 | 5, 6 | sylibr 234 | . 2 ⊢ (𝑇 ∈ Tarski → {𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇) |
| 8 | tskwe 9990 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ {𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇) → 𝑇 ∈ dom card) | |
| 9 | 7, 8 | mpdan 687 | 1 ⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 dom cdm 5685 ≺ csdm 8984 cardccrd 9975 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-card 9979 df-tsk 10789 |
| This theorem is referenced by: tskurn 10829 inaprc 10876 |
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