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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 3465 | . 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V) | |
2 | fvex 6859 | . . . 4 ⊢ (cf‘𝐴) ∈ V | |
3 | eleq1 2822 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | mpbii 232 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V) |
5 | 4 | 3ad2ant2 1135 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V) |
6 | neeq1 3003 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
7 | fveq2 6846 | . . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴)) | |
8 | eqeq12 2750 | . . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 687 | . . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) |
10 | breq2 5113 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴)) | |
11 | 10 | raleqbi1dv 3306 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
12 | 6, 9, 11 | 3anbi123d 1437 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))) |
13 | df-ina 10629 | . . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)} | |
14 | 12, 13 | elab2g 3636 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))) |
15 | 1, 5, 14 | pm5.21nii 380 | 1 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 Vcvv 3447 ∅c0 4286 𝒫 cpw 4564 class class class wbr 5109 ‘cfv 6500 ≺ csdm 8888 cfccf 9881 Inacccina 10627 |
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-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ina 10629 |
This theorem is referenced by: inawina 10634 omina 10635 gchina 10643 inar1 10719 inatsk 10722 tskcard 10725 tskuni 10727 gruina 10762 grur1 10764 |
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