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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-om | GIF version |
Description: A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-om | ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omind 13132 | . . . 4 ⊢ Ind ω | |
2 | bj-indeq 13127 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝐴 ↔ Ind ω)) | |
3 | 1, 2 | mpbiri 167 | . . 3 ⊢ (𝐴 = ω → Ind 𝐴) |
4 | vex 2689 | . . . . . 6 ⊢ 𝑥 ∈ V | |
5 | bj-omssind 13133 | . . . . . 6 ⊢ (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥)) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (Ind 𝑥 → ω ⊆ 𝑥) |
7 | sseq1 3120 | . . . . 5 ⊢ (𝐴 = ω → (𝐴 ⊆ 𝑥 ↔ ω ⊆ 𝑥)) | |
8 | 6, 7 | syl5ibr 155 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
9 | 8 | alrimiv 1846 | . . 3 ⊢ (𝐴 = ω → ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
10 | 3, 9 | jca 304 | . 2 ⊢ (𝐴 = ω → (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))) |
11 | bj-ssom 13134 | . . . . . . 7 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) | |
12 | 11 | biimpi 119 | . . . . . 6 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) → 𝐴 ⊆ ω) |
13 | 12 | adantl 275 | . . . . 5 ⊢ ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ ω) |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ ω)) |
15 | bj-omssind 13133 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) | |
16 | 15 | adantrd 277 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → ω ⊆ 𝐴)) |
17 | 14, 16 | jcad 305 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → (𝐴 ⊆ ω ∧ ω ⊆ 𝐴))) |
18 | eqss 3112 | . . 3 ⊢ (𝐴 = ω ↔ (𝐴 ⊆ ω ∧ ω ⊆ 𝐴)) | |
19 | 17, 18 | syl6ibr 161 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 = ω)) |
20 | 10, 19 | impbid2 142 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ωcom 4504 Ind wind 13124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 ax-pr 4131 ax-un 4355 ax-bd0 13011 ax-bdor 13014 ax-bdex 13017 ax-bdeq 13018 ax-bdel 13019 ax-bdsb 13020 ax-bdsep 13082 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: bj-2inf 13136 bj-inf2vn 13172 bj-inf2vn2 13173 |
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