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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 3450 | . 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V) | |
2 | fvex 6543 | . . . 4 ⊢ (cf‘𝐴) ∈ V | |
3 | eleq1 2868 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | mpbii 234 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V) |
5 | 4 | 3ad2ant2 1125 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V) |
6 | neeq1 3044 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
7 | fveq2 6530 | . . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴)) | |
8 | eqeq12 2806 | . . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 684 | . . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴)) |
10 | breq2 4960 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴)) | |
11 | 10 | raleqbi1dv 3360 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) |
12 | 6, 9, 11 | 3anbi123d 1426 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))) |
13 | df-ina 9942 | . . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)} | |
14 | 12, 13 | elab2g 3602 | . 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 207 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∀wral 3103 Vcvv 3432 ∅c0 4206 𝒫 cpw 4447 class class class wbr 4956 ‘cfv 6217 ≺ csdm 8346 cfccf 9201 Inacccina 9940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 ax-nul 5095 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-iota 6181 df-fv 6225 df-ina 9942 |
This theorem is referenced by: inawina 9947 omina 9948 gchina 9956 inar1 10032 inatsk 10035 tskcard 10038 tskuni 10040 gruina 10075 grur1 10077 |
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