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Theorem bj-omssind 10059
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 2178 . . 3 𝑥𝐴
2 nfv 1421 . . 3 𝑥Ind 𝐴
3 bj-indeq 10053 . . . 4 (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
43biimprd 147 . . 3 (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
51, 2, 4bj-intabssel1 9929 . 2 (𝐴𝑉 → (Ind 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6 bj-dfom 10057 . . 3 ω = {𝑥 ∣ Ind 𝑥}
76sseq1i 2969 . 2 (ω ⊆ 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
85, 7syl6ibr 151 1 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  {cab 2026  wss 2917   cint 3615  ωcom 4313  Ind wind 10050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-iom 4314  df-bj-ind 10051
This theorem is referenced by:  bj-om  10061  peano5set  10064
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