Theorem bj-omex2 14768

Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 14732 (see bj-2inf 14729 for the equivalence of the latter with bj-omex 14733). (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 14767	. . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
2		vex 2742	. . . 4 ⊢ 𝑎 ∈ V
3		bdcv 14639	. . . . 5 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 14765	. . . 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 1602	. 2 ⊢ ∃𝑎 𝑎 = ω
7	6	issetri 2748	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 708  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  Vcvv 2739  ∅c0 3424  suc csuc 4367  ωcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14604  ax-bdim 14605  ax-bdor 14607  ax-bdex 14610  ax-bdeq 14611  ax-bdel 14612  ax-bdsb 14613  ax-bdsep 14675  ax-bdsetind 14759  ax-inf2 14767

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14632  df-bj-ind 14718

This theorem is referenced by: (None)