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Theorem bj-omssind 13060
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 2258 . . 3 𝑥𝐴
2 nfv 1493 . . 3 𝑥Ind 𝐴
3 bj-indeq 13054 . . . 4 (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
43biimprd 157 . . 3 (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
51, 2, 4bj-intabssel1 12924 . 2 (𝐴𝑉 → (Ind 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6 bj-dfom 13058 . . 3 ω = {𝑥 ∣ Ind 𝑥}
76sseq1i 3093 . 2 (ω ⊆ 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
85, 7syl6ibr 161 1 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  {cab 2103  wss 3041   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-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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
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-v 2662  df-in 3047  df-ss 3054  df-int 3742  df-iom 4475  df-bj-ind 13052
This theorem is referenced by:  bj-om  13062  peano5set  13065
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