Theorem bj-inf2vn2 14888

Description: A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 14887. (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 14883	. . 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 1455	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)))
4		df-ral 2460	. . . . . 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 14886	. . . . 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 1874	. . 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 14850	. 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 708  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3131  ∅c0 3424  suc csuc 4367  ωcom 4591  Ind wind 14839

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-setind 4538  ax-bd0 14726  ax-bdor 14729  ax-bdex 14732  ax-bdeq 14733  ax-bdel 14734  ax-bdsb 14735  ax-bdsep 14797

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 14754  df-bj-ind 14840

This theorem is referenced by: (None)