Theorem bj-inf2vnlem4 16336

Description: Lemma for bj-inf2vn2 16338. (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 16334	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))
2		nfv 1574	. . . 4 ⊢ Ⅎ𝑧(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
3		nfv 1574	. . . 4 ⊢ Ⅎ𝑧(𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)
4		nfv 1574	. . . 4 ⊢ Ⅎ𝑢(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)
5		nfv 1574	. . . 4 ⊢ Ⅎ𝑢(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
6		eleq1 2292	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
7		eleq1 2292	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑍 ↔ 𝑡 ∈ 𝑍))
8	6, 7	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
9	8	biimpd 144	. . . 4 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) → (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
10		eleq1 2292	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
11		eleq1 2292	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝑍 ↔ 𝑢 ∈ 𝑍))
12	10, 11	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑢 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
13	12	biimprd 158	. . . 4 ⊢ (𝑧 = 𝑢 → ((𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)))
14	2, 3, 4, 5, 9, 13	setindis 16330	. . 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		ssalel 3212	. 2 ⊢ (𝐴 ⊆ 𝑍 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍))
17	15, 16	imbitrrdi 162	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))

Syntax hints:   → wi 4   ∨ wo 713  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3197  ∅c0 3491  suc csuc 4456  Ind wind 16289

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-suc 4462  df-bj-ind 16290

This theorem is referenced by:  bj-inf2vn2  16338