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Mirrors > Home > MPE Home > Th. List > elina | Structured version Visualization version GIF version |
Description: Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.) |
Ref | Expression |
---|---|
elina | ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3482 | . 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V) | |
2 | fvex 6914 | . . . 4 ⊢ (cf‘𝐴) ∈ V | |
3 | eleq1 2814 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | mpbii 232 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V) |
5 | 4 | 3ad2ant2 1131 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V) |
6 | neeq1 2993 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
7 | fveq2 6901 | . . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴)) | |
8 | eqeq12 2743 | . . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 686 | . . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) |
10 | breq2 5157 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴)) | |
11 | 10 | raleqbi1dv 3323 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
12 | 6, 9, 11 | 3anbi123d 1433 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))) |
13 | df-ina 10728 | . . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)} | |
14 | 12, 13 | elab2g 3668 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))) |
15 | 1, 5, 14 | pm5.21nii 377 | 1 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 Vcvv 3462 ∅c0 4325 𝒫 cpw 4607 class class class wbr 5153 ‘cfv 6554 ≺ csdm 8973 cfccf 9980 Inacccina 10726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-iota 6506 df-fv 6562 df-ina 10728 |
This theorem is referenced by: inawina 10733 omina 10734 gchina 10742 inar1 10818 inatsk 10821 tskcard 10824 tskuni 10826 gruina 10861 grur1 10863 |
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