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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-2inf | GIF version |
Description: Two formulations of the axiom of infinity (see ax-infvn 11177 and bj-omex 11178) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-2inf | ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2083 | . . . 4 ⊢ ω = ω | |
2 | bj-om 11173 | . . . 4 ⊢ (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))) | |
3 | 1, 2 | mpbii 146 | . . 3 ⊢ (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))) |
4 | bj-indeq 11165 | . . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω)) | |
5 | sseq1 3031 | . . . . . . 7 ⊢ (𝑥 = ω → (𝑥 ⊆ 𝑦 ↔ ω ⊆ 𝑦)) | |
6 | 5 | imbi2d 228 | . . . . . 6 ⊢ (𝑥 = ω → ((Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦))) |
7 | 6 | albidv 1747 | . . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))) |
8 | 4, 7 | anbi12d 457 | . . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))) |
9 | 8 | spcegv 2697 | . . 3 ⊢ (ω ∈ V → ((Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)) → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))) |
10 | 3, 9 | mpd 13 | . 2 ⊢ (ω ∈ V → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
11 | vex 2615 | . . . . . 6 ⊢ 𝑥 ∈ V | |
12 | bj-om 11173 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))) | |
13 | 11, 12 | ax-mp 7 | . . . . 5 ⊢ (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
14 | 13 | biimpri 131 | . . . 4 ⊢ ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → 𝑥 = ω) |
15 | 14 | eximi 1532 | . . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω) |
16 | isset 2616 | . . 3 ⊢ (ω ∈ V ↔ ∃𝑥 𝑥 = ω) | |
17 | 15, 16 | sylibr 132 | . 2 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ω ∈ V) |
18 | 10, 17 | impbii 124 | 1 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1283 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 ωcom 4367 Ind wind 11162 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-nul 3930 ax-pr 3999 ax-un 4223 ax-bd0 11045 ax-bdor 11048 ax-bdex 11051 ax-bdeq 11052 ax-bdel 11053 ax-bdsb 11054 ax-bdsep 11116 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-sn 3428 df-pr 3429 df-uni 3628 df-int 3663 df-suc 4161 df-iom 4368 df-bdc 11073 df-bj-ind 11163 |
This theorem is referenced by: bj-omex 11178 |
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