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Theorem bj-omssind 8511
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 2154 . . 3 xA
2 nfv 1397 . . 3 xInd A
3 bj-indeq 8506 . . . 4 (x = A → (Ind x ↔ Ind A))
43biimprd 147 . . 3 (x = A → (Ind A → Ind x))
51, 2, 4bj-intabssel1 8386 . 2 (A 𝑉 → (Ind A {x ∣ Ind x} ⊆ A))
6 bj-dfom 8509 . . 3 𝜔 = {x ∣ Ind x}
76sseq1i 2940 . 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 1226   wcel 1369  {cab 2002  wss 2888   cint 3581  𝜔com 4231  Ind wind 8503
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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ral 2283  df-v 2531  df-in 2895  df-ss 2902  df-int 3582  df-iom 4232  df-bj-ind 8504
This theorem is referenced by:  bj-om  8513
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