Theorem bj-inf2vnlem4 12134

Description: Lemma for bj-inf2vn2 12136. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vnlem4	⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑍,𝑦

Proof of Theorem bj-inf2vnlem4

Dummy variables 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem2 12132	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))
2		nfv 1467	. . . 4 ⊢ Ⅎ𝑧(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
3		nfv 1467	. . . 4 ⊢ Ⅎ𝑧(𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)
4		nfv 1467	. . . 4 ⊢ Ⅎ𝑢(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)
5		nfv 1467	. . . 4 ⊢ Ⅎ𝑢(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
6		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
7		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑍 ↔ 𝑡 ∈ 𝑍))
8	6, 7	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
9	8	biimpd 143	. . . 4 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) → (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
10		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
11		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝑍 ↔ 𝑢 ∈ 𝑍))
12	10, 11	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝑢 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
13	12	biimprd 157	. . . 4 ⊢ (𝑧 = 𝑢 → ((𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)))
14	2, 3, 4, 5, 9, 13	setindis 12128	. . 3 ⊢ (∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)) → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍))
15	1, 14	syl6 33	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)))
16		dfss2 3015	. 2 ⊢ (𝐴 ⊆ 𝑍 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍))
17	15, 16	syl6ibr 161	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))

Syntax hints:   → wi 4   ∨ wo 665  ∀wal 1288   = wceq 1290   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361   ⊆ wss 3000  ∅c0 3287  suc csuc 4201  Ind wind 12087

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-suc 4207  df-bj-ind 12088

This theorem is referenced by:  bj-inf2vn2  12136