Theorem bj-omex2 11872

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 11836 (see bj-2inf 11833 for the equivalence of the latter with bj-omex 11837). (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 11871	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2622	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 11739	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 11869	. . . 4 ⊢ (𝑎 ∈ V → (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω))
5	2, 4	ax-mp 7	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω)
6	1, 5	eximii 1538	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2628	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 103   ∨ wo 664  ∀wal 1287   = wceq 1289   ∈ wcel 1438  ∃wrex 2360  Vcvv 2619  ∅c0 3286  suc csuc 4192  ωcom 4405

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965  ax-pr 4036  ax-un 4260  ax-bd0 11704  ax-bdim 11705  ax-bdor 11707  ax-bdex 11710  ax-bdeq 11711  ax-bdel 11712  ax-bdsb 11713  ax-bdsep 11775  ax-bdsetind 11863  ax-inf2 11871

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-suc 4198  df-iom 4406  df-bdc 11732  df-bj-ind 11822

This theorem is referenced by: (None)