Theorem bj-inf2vnlem3 17164

Description: Lemma for bj-inf2vn 17166. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-inf2vnlem3.bd1	⊢ BOUNDED 𝐴
bj-inf2vnlem3.bd2	⊢ BOUNDED 𝑍

Assertion

Ref	Expression
bj-inf2vnlem3	⊢ (∀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-inf2vnlem3

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

Step	Hyp	Ref	Expression
1		bj-inf2vnlem2 17163	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))
2		bj-inf2vnlem3.bd1	. . . . . 6 ⊢ BOUNDED 𝐴
3	2	bdeli 17038	. . . . 5 ⊢ BOUNDED 𝑧 ∈ 𝐴
4		bj-inf2vnlem3.bd2	. . . . . 6 ⊢ BOUNDED 𝑍
5	4	bdeli 17038	. . . . 5 ⊢ BOUNDED 𝑧 ∈ 𝑍
6	3, 5	ax-bdim 17006	. . . 4 ⊢ BOUNDED (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)
7		nfv 1581	. . . 4 ⊢ Ⅎ𝑧(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
8		nfv 1581	. . . 4 ⊢ Ⅎ𝑧(𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)
9		nfv 1581	. . . 4 ⊢ Ⅎ𝑢(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)
10		nfv 1581	. . . 4 ⊢ Ⅎ𝑢(𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)
11		eleq1 2301	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
12		eleq1 2301	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑍 ↔ 𝑡 ∈ 𝑍))
13	11, 12	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
14	13	biimpd 144	. . . 4 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) → (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍)))
15		eleq1 2301	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
16		eleq1 2301	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝑍 ↔ 𝑢 ∈ 𝑍))
17	15, 16	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑢 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍) ↔ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
18	17	biimprd 158	. . . 4 ⊢ (𝑧 = 𝑢 → ((𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)))
19	6, 7, 8, 9, 10, 14, 18	bdsetindis 17161	. . 3 ⊢ (∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)) → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍))
20	1, 19	syl6 33	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍)))
21		ssalel 3235	. 2 ⊢ (𝐴 ⊆ 𝑍 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑍))
22	20, 21	imbitrrdi 162	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))

This proof depends on syntax axioms:   → wi 4   ∨ wo 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ⊆ wss 3220  ∅c0 3520  suc csuc 4510  BOUNDED wbdc 17032  Ind wind 17118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bdim 17006  ax-bdsetind 17160

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-suc 4516  df-bdc 17033  df-bj-ind 17119

This theorem is used by:  bj-inf2vn  17166