Theorem bj-inf2vn 16856

Description: A sufficient condition for ω to be a set. See bj-inf2vn2 16857 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-inf2vn.1	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bj-inf2vn	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem bj-inf2vn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem1 16852	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
2		biimp 118	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (𝑥 ∈ 𝐴 → (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)))
3	2	alimi 1504	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)))
4		df-ral 2527	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)))
5	3, 4	sylibr 134	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦))
6		bj-inf2vn.1	. . . . . 6 ⊢ BOUNDED 𝐴
7		bdcv 16730	. . . . . 6 ⊢ BOUNDED 𝑧
8	6, 7	bj-inf2vnlem3 16854	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑧 → 𝐴 ⊆ 𝑧))
9	5, 8	syl 14	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧 → 𝐴 ⊆ 𝑧))
10	9	alrimiv 1923	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧))
11	1, 10	jca 306	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧)))
12		bj-om 16819	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧))))
13	11, 12	imbitrrid 156	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523   ⊆ wss 3214  ∅c0 3512  suc csuc 4491  ωcom 4717  BOUNDED wbdc 16722  Ind wind 16808

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 2207  ax-14 2208  ax-ext 2216  ax-nul 4241  ax-pr 4327  ax-un 4559  ax-bd0 16695  ax-bdim 16696  ax-bdor 16698  ax-bdex 16701  ax-bdeq 16702  ax-bdel 16703  ax-bdsb 16704  ax-bdsep 16766  ax-bdsetind 16850

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-suc 4497  df-iom 4718  df-bdc 16723  df-bj-ind 16809

This theorem is referenced by:  bj-omex2  16859  bj-nn0sucALT  16860