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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 4566 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑇 → 𝑦 ⊆ 𝑇) | |
2 | tskssel 10627 | . . . . . 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 3141 | . . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑦 ∈ 𝒫 𝑇(𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇)) |
6 | rabss 4028 | . . 3 ⊢ ({𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝒫 𝑇(𝑦 ≺ 𝑇 → 𝑦 ∈ 𝑇)) | |
7 | 5, 6 | sylibr 233 | . 2 ⊢ (𝑇 ∈ Tarski → {𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇) |
8 | tskwe 9820 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ {𝑦 ∈ 𝒫 𝑇 ∣ 𝑦 ≺ 𝑇} ⊆ 𝑇) → 𝑇 ∈ dom card) | |
9 | 7, 8 | mpdan 686 | 1 ⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3063 {crab 3406 ⊆ wss 3909 𝒫 cpw 4559 class class class wbr 5104 dom cdm 5631 ≺ csdm 8816 cardccrd 9805 Tarskictsk 10618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 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-ord 6317 df-on 6318 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-card 9809 df-tsk 10619 |
This theorem is referenced by: tskurn 10659 inaprc 10706 |
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