Theorem bj-omex2 16717

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 16681 (see bj-2inf 16678 for the equivalence of the latter with bj-omex 16682). (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 16716	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2815	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 16588	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 16714	. . . 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 1651	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2822	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  Vcvv 2812  ∅c0 3505  suc csuc 4477  ωcom 4703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4229  ax-pr 4314  ax-un 4545  ax-bd0 16553  ax-bdim 16554  ax-bdor 16556  ax-bdex 16559  ax-bdeq 16560  ax-bdel 16561  ax-bdsb 16562  ax-bdsep 16624  ax-bdsetind 16708  ax-inf2 16716

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-sn 3688  df-pr 3689  df-uni 3908  df-int 3943  df-suc 4483  df-iom 4704  df-bdc 16581  df-bj-ind 16667

This theorem is referenced by: (None)