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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 13303 | . . . 4 ⊢ Ind ω | |
2 | bj-indeq 13298 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝐴 ↔ Ind ω)) | |
3 | 1, 2 | mpbiri 167 | . . 3 ⊢ (𝐴 = ω → Ind 𝐴) |
4 | vex 2692 | . . . . . 6 ⊢ 𝑥 ∈ V | |
5 | bj-omssind 13304 | . . . . . 6 ⊢ (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥)) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (Ind 𝑥 → ω ⊆ 𝑥) |
7 | sseq1 3125 | . . . . 5 ⊢ (𝐴 = ω → (𝐴 ⊆ 𝑥 ↔ ω ⊆ 𝑥)) | |
8 | 6, 7 | syl5ibr 155 | . . . 4 ⊢ (𝐴 = ω → (Ind 𝑥 → 𝐴 ⊆ 𝑥)) |
9 | 8 | alrimiv 1847 | . . 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 13305 | . . . . . . 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 13304 | . . . . 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 3117 | . . 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 1330 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 ωcom 4512 Ind wind 13295 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-bd0 13182 ax-bdor 13185 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: bj-2inf 13307 bj-inf2vn 13343 bj-inf2vn2 13344 |
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