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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 10058 | . . . 4 ⊢ Ind ω | |
2 | bj-indeq 10053 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝐴 ↔ Ind ω)) | |
3 | 1, 2 | mpbiri 157 | . . 3 ⊢ (𝐴 = ω → Ind 𝐴) |
4 | vex 2560 | . . . . . 6 ⊢ 𝑥 ∈ V | |
5 | bj-omssind 10059 | . . . . . 6 ⊢ (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥)) | |
6 | 4, 5 | ax-mp 7 | . . . . 5 ⊢ (Ind 𝑥 → ω ⊆ 𝑥) |
7 | sseq1 2966 | . . . . 5 ⊢ (𝐴 = ω → (𝐴 ⊆ 𝑥 ↔ ω ⊆ 𝑥)) | |
8 | 6, 7 | syl5ibr 145 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
9 | 8 | alrimiv 1754 | . . 3 ⊢ (𝐴 = ω → ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
10 | 3, 9 | jca 290 | . 2 ⊢ (𝐴 = ω → (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))) |
11 | bj-ssom 10060 | . . . . . . 7 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) | |
12 | 11 | biimpi 113 | . . . . . 6 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) → 𝐴 ⊆ ω) |
13 | 12 | adantl 262 | . . . . 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 10059 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) | |
16 | 15 | adantrd 264 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → ω ⊆ 𝐴)) |
17 | 14, 16 | jcad 291 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → (𝐴 ⊆ ω ∧ ω ⊆ 𝐴))) |
18 | eqss 2960 | . . 3 ⊢ (𝐴 = ω ↔ (𝐴 ⊆ ω ∧ ω ⊆ 𝐴)) | |
19 | 17, 18 | syl6ibr 151 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 = ω)) |
20 | 10, 19 | impbid2 131 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 ωcom 4313 Ind wind 10050 |
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-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-2inf 10062 bj-inf2vn 10099 bj-inf2vn2 10100 |
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