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Theorem bj-omex2 9407
Description: Using bounded set induction and the strong axiom of infinity, 𝜔 is a set, that is, we recover ax-infvn 9375 (see bj-2inf 9372 for the equivalence of the latter with bj-omex 9376). (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 9406 . . 3 𝑎x(x 𝑎 ↔ (x = ∅ y 𝑎 x = suc y))
2 vex 2554 . . . 4 𝑎 V
3 bdcv 9283 . . . . 5 BOUNDED 𝑎
43bj-inf2vn 9404 . . . 4 (𝑎 V → (x(x 𝑎 ↔ (x = ∅ y 𝑎 x = suc y)) → 𝑎 = 𝜔))
52, 4ax-mp 7 . . 3 (x(x 𝑎 ↔ (x = ∅ y 𝑎 x = suc y)) → 𝑎 = 𝜔)
61, 5eximii 1490 . 2 𝑎 𝑎 = 𝜔
76issetri 2558 1 𝜔 V
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 628  wal 1240   = wceq 1242   wcel 1390  wrex 2301  Vcvv 2551  c0 3218  suc csuc 4068  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9248  ax-bdim 9249  ax-bdor 9251  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319  ax-bdsetind 9398  ax-inf2 9406
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by: (None)
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