Theorem bj-inf2vn 15947

Description: A sufficient condition for ω to be a set. See bj-inf2vn2 15948 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 15943	. . 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 1478	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)))
4		df-ral 2489	. . . . . 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 15821	. . . . . 6 ⊢ BOUNDED 𝑧
8	6, 7	bj-inf2vnlem3 15945	. . . . 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 1897	. . 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 15910	. 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 710  ∀wal 1371   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485   ⊆ wss 3166  ∅c0 3460  suc csuc 4413  ωcom 4639  BOUNDED wbdc 15813  Ind wind 15899

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4171  ax-pr 4254  ax-un 4481  ax-bd0 15786  ax-bdim 15787  ax-bdor 15789  ax-bdex 15792  ax-bdeq 15793  ax-bdel 15794  ax-bdsb 15795  ax-bdsep 15857  ax-bdsetind 15941

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4419  df-iom 4640  df-bdc 15814  df-bj-ind 15900

This theorem is referenced by:  bj-omex2  15950  bj-nn0sucALT  15951