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Theorem bj-omssind 11022
Description: ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-omssind (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Proof of Theorem bj-omssind
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2223 . . 3 𝑥𝐴
2 nfv 1462 . . 3 𝑥Ind 𝐴
3 bj-indeq 11016 . . . 4 (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
43biimprd 156 . . 3 (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
51, 2, 4bj-intabssel1 10885 . 2 (𝐴𝑉 → (Ind 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6 bj-dfom 11020 . . 3 ω = {𝑥 ∣ Ind 𝑥}
76sseq1i 3033 . 2 (ω ⊆ 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
85, 7syl6ibr 160 1 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  {cab 2069  wss 2983   cint 3657  ωcom 4360  Ind wind 11013
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-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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
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-v 2613  df-in 2989  df-ss 2996  df-int 3658  df-iom 4361  df-bj-ind 11014
This theorem is referenced by:  bj-om  11024  peano5set  11027
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