Theorem bj-omex2 17015

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 16979 (see bj-2inf 16976 for the equivalence of the latter with bj-omex 16980). (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.

Step	Hyp	Ref	Expression
1		ax-inf2 17014	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2824	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 16886	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 17012	. . . 4 ⊢ (𝑎 ∈ V → (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω))
5	2, 4	ax-mp 5	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω)
6	1, 5	eximii 1655	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2831	1 ⊢ ω ∈ V

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∨ wo 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ∅c0 3520  suc csuc 4510  ωcom 4737

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4259  ax-pr 4346  ax-un 4578  ax-bd0 16851  ax-bdim 16852  ax-bdor 16854  ax-bdex 16857  ax-bdeq 16858  ax-bdel 16859  ax-bdsep 16922  ax-bdsetind 17006  ax-inf2 17014

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-suc 4516  df-iom 4738  df-bdc 16879  df-bj-ind 16965

This theorem is used by: (None)