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Theorem bj-omssind 10418
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 2194 . . 3 𝑥𝐴
2 nfv 1437 . . 3 𝑥Ind 𝐴
3 bj-indeq 10412 . . . 4 (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
43biimprd 151 . . 3 (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
51, 2, 4bj-intabssel1 10288 . 2 (𝐴𝑉 → (Ind 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6 bj-dfom 10416 . . 3 ω = {𝑥 ∣ Ind 𝑥}
76sseq1i 2996 . 2 (ω ⊆ 𝐴 {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
85, 7syl6ibr 155 1 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  {cab 2042  wss 2944   cint 3642  ωcom 4340  Ind wind 10409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2951  df-ss 2958  df-int 3643  df-iom 4341  df-bj-ind 10410
This theorem is referenced by:  bj-om  10420  peano5set  10423
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