Theorem bj-omex2 13185

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13149 (see bj-2inf 13146 for the equivalence of the latter with bj-omex 13150). (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 13184	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2689	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 13056	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 13182	. . . 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 1581	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2695	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686  ∅c0 3363  suc csuc 4287  ωcom 4504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13021  ax-bdim 13022  ax-bdor 13024  ax-bdex 13027  ax-bdeq 13028  ax-bdel 13029  ax-bdsb 13030  ax-bdsep 13092  ax-bdsetind 13176  ax-inf2 13184

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13049  df-bj-ind 13135

This theorem is referenced by: (None)