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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omind | GIF version |
Description: ω is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-omind | ⊢ Ind ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-indint 13056 | . 2 ⊢ Ind ∩ {𝑥 ∈ V ∣ Ind 𝑥} | |
2 | bj-dfom 13058 | . . . 4 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} | |
3 | rabab 2681 | . . . . 5 ⊢ {𝑥 ∈ V ∣ Ind 𝑥} = {𝑥 ∣ Ind 𝑥} | |
4 | 3 | inteqi 3745 | . . . 4 ⊢ ∩ {𝑥 ∈ V ∣ Ind 𝑥} = ∩ {𝑥 ∣ Ind 𝑥} |
5 | 2, 4 | eqtr4i 2141 | . . 3 ⊢ ω = ∩ {𝑥 ∈ V ∣ Ind 𝑥} |
6 | bj-indeq 13054 | . . 3 ⊢ (ω = ∩ {𝑥 ∈ V ∣ Ind 𝑥} → (Ind ω ↔ Ind ∩ {𝑥 ∈ V ∣ Ind 𝑥})) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (Ind ω ↔ Ind ∩ {𝑥 ∈ V ∣ Ind 𝑥}) |
8 | 1, 7 | mpbir 145 | 1 ⊢ Ind ω |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 {cab 2103 {crab 2397 Vcvv 2660 ∩ cint 3741 ωcom 4474 Ind wind 13051 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 ax-pr 4101 ax-un 4325 ax-bd0 12938 ax-bdor 12941 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-nul 3334 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: bj-om 13062 bj-peano2 13064 peano5set 13065 |
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