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Theorem bj-omssind 11268
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 2225 . . 3 𝑥𝐴
2 nfv 1464 . . 3 𝑥Ind 𝐴
3 bj-indeq 11262 . . . 4 (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
43biimprd 156 . . 3 (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
51, 2, 4bj-intabssel1 11128 . 2 (𝐴𝑉 → (Ind 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6 bj-dfom 11266 . . 3 ω = {𝑥 ∣ Ind 𝑥}
76sseq1i 3039 . 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 1287  wcel 1436  {cab 2071  wss 2988   cint 3671  ωcom 4378  Ind wind 11259
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-in 2994  df-ss 3001  df-int 3672  df-iom 4379  df-bj-ind 11260
This theorem is referenced by:  bj-om  11270  peano5set  11273
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