Theorem bj-inf2vn2 16745

Description: A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 16744. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vn2	⊢ (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-inf2vn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem1 16740	. . 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 2525	. . . . . 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-inf2vnlem4 16743	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑧 → 𝐴 ⊆ 𝑧))
7	5, 6	syl 14	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧 → 𝐴 ⊆ 𝑧))
8	7	alrimiv 1923	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧))
9	1, 8	jca 306	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧)))
10		bj-om 16707	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧 → 𝐴 ⊆ 𝑧))))
11	9, 10	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 2203  ∀wral 2520  ∃wrex 2521   ⊆ wss 3211  ∅c0 3508  suc csuc 4486  ωcom 4712  Ind wind 16696

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 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-bd0 16583  ax-bdor 16586  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by: (None)