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

Theorem bj-omind 7156
Description: 𝜔 is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-omind Ind 𝜔

Proof of Theorem bj-omind
StepHypRef Expression
1 bj-indint 7154 . 2 Ind {x V ∣ Ind x}
2 bj-dfom 7155 . . . 4 𝜔 = {x ∣ Ind x}
3 rabab 2552 . . . . 5 {x V ∣ Ind x} = {x ∣ Ind x}
43inteqi 3593 . . . 4 {x V ∣ Ind x} = {x ∣ Ind x}
52, 4eqtr4i 2045 . . 3 𝜔 = {x V ∣ Ind x}
6 bj-indeq 7152 . . 3 (𝜔 = {x V ∣ Ind x} → (Ind 𝜔 ↔ Ind {x V ∣ Ind x}))
75, 6ax-mp 7 . 2 (Ind 𝜔 ↔ Ind {x V ∣ Ind x})
81, 7mpbir 134 1 Ind 𝜔
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228  {cab 2008  {crab 2288  Vcvv 2535   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-in1 532  ax-in2 533  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-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111
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-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150
This theorem is referenced by:  bj-om  7159  bj-peano2  7161
  Copyright terms: Public domain W3C validator