Theorem bj-omex2 16575

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 16539 (see bj-2inf 16536 for the equivalence of the latter with bj-omex 16540). (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 16574	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2805	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 16446	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 16572	. . . 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 1650	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2812	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 715  ∀wal 1395   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802  ∅c0 3494  suc csuc 4462  ωcom 4688

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16411  ax-bdim 16412  ax-bdor 16414  ax-bdex 16417  ax-bdeq 16418  ax-bdel 16419  ax-bdsb 16420  ax-bdsep 16482  ax-bdsetind 16566  ax-inf2 16574

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16439  df-bj-ind 16525

This theorem is referenced by: (None)