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Theorem bj-omex2 13102
Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13066 (see bj-2inf 13063 for the equivalence of the latter with bj-omex 13067). (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 13101 . . 3 𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))
2 vex 2663 . . . 4 𝑎 ∈ V
3 bdcv 12973 . . . . 5 BOUNDED 𝑎
43bj-inf2vn 13099 . . . 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 1566 . 2 𝑎 𝑎 = ω
76issetri 2669 1 ω ∈ V
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 682  wal 1314   = wceq 1316  wcel 1465  wrex 2394  Vcvv 2660  c0 3333  suc csuc 4257  ωcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdim 12939  ax-bdor 12941  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-bdsetind 13093  ax-inf2 13101
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by: (None)
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