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Theorem bj-omssind 9392
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 (A 𝑉 → (Ind A → 𝜔 ⊆ A))

Proof of Theorem bj-omssind
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 nfcv 2175 . . 3 xA
2 nfv 1418 . . 3 xInd A
3 bj-indeq 9386 . . . 4 (x = A → (Ind x ↔ Ind A))
43biimprd 147 . . 3 (x = A → (Ind A → Ind x))
51, 2, 4bj-intabssel1 9264 . 2 (A 𝑉 → (Ind A {x ∣ Ind x} ⊆ A))
6 bj-dfom 9390 . . 3 𝜔 = {x ∣ Ind x}
76sseq1i 2963 . 2 (𝜔 ⊆ A {x ∣ Ind x} ⊆ A)
85, 7syl6ibr 151 1 (A 𝑉 → (Ind A → 𝜔 ⊆ A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {cab 2023  wss 2911   cint 3606  𝜔com 4256  Ind wind 9383
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-iom 4257  df-bj-ind 9384
This theorem is referenced by:  bj-om  9394  peano5set  9397
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