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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-ssom | GIF version |
Description: A characterization of subclasses of 𝜔. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ssom | ⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ 𝜔) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 3622 | . . 3 ⊢ (A ⊆ ∩ {y ∣ Ind y} ↔ ∀x ∈ {y ∣ Ind y}A ⊆ x) | |
2 | df-ral 2305 | . . 3 ⊢ (∀x ∈ {y ∣ Ind y}A ⊆ x ↔ ∀x(x ∈ {y ∣ Ind y} → A ⊆ x)) | |
3 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
4 | bj-indeq 9388 | . . . . . 6 ⊢ (y = x → (Ind y ↔ Ind x)) | |
5 | 3, 4 | elab 2681 | . . . . 5 ⊢ (x ∈ {y ∣ Ind y} ↔ Ind x) |
6 | 5 | imbi1i 227 | . . . 4 ⊢ ((x ∈ {y ∣ Ind y} → A ⊆ x) ↔ (Ind x → A ⊆ x)) |
7 | 6 | albii 1356 | . . 3 ⊢ (∀x(x ∈ {y ∣ Ind y} → A ⊆ x) ↔ ∀x(Ind x → A ⊆ x)) |
8 | 1, 2, 7 | 3bitrri 196 | . 2 ⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ ∩ {y ∣ Ind y}) |
9 | bj-dfom 9392 | . . . 4 ⊢ 𝜔 = ∩ {y ∣ Ind y} | |
10 | 9 | eqcomi 2041 | . . 3 ⊢ ∩ {y ∣ Ind y} = 𝜔 |
11 | 10 | sseq2i 2964 | . 2 ⊢ (A ⊆ ∩ {y ∣ Ind y} ↔ A ⊆ 𝜔) |
12 | 8, 11 | bitri 173 | 1 ⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ 𝜔) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 {cab 2023 ∀wral 2300 ⊆ wss 2911 ∩ cint 3606 𝜔com 4256 Ind wind 9385 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-in 2918 df-ss 2925 df-int 3607 df-iom 4257 df-bj-ind 9386 |
This theorem is referenced by: bj-om 9396 |
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