 Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omssind Structured version   GIF version

Theorem bj-omssind 7157
 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 2160 . . 3 xA
2 nfv 1402 . . 3 xInd A
3 bj-indeq 7152 . . . 4 (x = A → (Ind x ↔ Ind A))
43biimprd 147 . . 3 (x = A → (Ind A → Ind x))
51, 2, 4bj-intabssel1 7036 . 2 (A 𝑉 → (Ind A {x ∣ Ind x} ⊆ A))
6 bj-dfom 7155 . . 3 𝜔 = {x ∣ Ind x}
76sseq1i 2946 . 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 1228   ∈ wcel 1374  {cab 2008   ⊆ wss 2894  ∩ cint 3589  𝜔com 4240  Ind wind 7149 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-int 3590  df-iom 4241  df-bj-ind 7150 This theorem is referenced by:  bj-om  7159
 Copyright terms: Public domain W3C validator