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Theorem bj-omex2 10930
 Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 10894 (see bj-2inf 10891 for the equivalence of the latter with bj-omex 10895). (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 10929 . . 3 𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))
2 vex 2605 . . . 4 𝑎 ∈ V
3 bdcv 10797 . . . . 5 BOUNDED 𝑎
43bj-inf2vn 10927 . . . 4 (𝑎 ∈ V → (∀𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω))
52, 4ax-mp 7 . . 3 (∀𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω)
61, 5eximii 1534 . 2 𝑎 𝑎 = ω
76issetri 2609 1 ω ∈ V
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   ∨ wo 662  ∀wal 1283   = wceq 1285   ∈ wcel 1434  ∃wrex 2350  Vcvv 2602  ∅c0 3258  suc csuc 4128  ωcom 4339 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10762  ax-bdim 10763  ax-bdor 10765  ax-bdex 10768  ax-bdeq 10769  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833  ax-bdsetind 10921  ax-inf2 10929 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10790  df-bj-ind 10880 This theorem is referenced by: (None)
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