Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omex2 GIF version

Theorem bj-omex2 13175
Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13139 (see bj-2inf 13136 for the equivalence of the latter with bj-omex 13140). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-omex2 ω ∈ V

Proof of Theorem bj-omex2
Dummy variables 𝑥 𝑦 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 13174 . . 3 𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))
2 vex 2689 . . . 4 𝑎 ∈ V
3 bdcv 13046 . . . . 5 BOUNDED 𝑎
43bj-inf2vn 13172 . . . 4 (𝑎 ∈ V → (∀𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω))
52, 4ax-mp 5 . . 3 (∀𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω)
61, 5eximii 1581 . 2 𝑎 𝑎 = ω
76issetri 2695 1 ω ∈ V
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 697  wal 1329   = wceq 1331  wcel 1480  wrex 2417  Vcvv 2686  c0 3363  suc csuc 4287  ωcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdim 13012  ax-bdor 13014  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-bdsetind 13166  ax-inf2 13174
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator