Theorem bj-inf2vnlem2 14006

Description: Lemma for bj-inf2vnlem3 14007 and bj-inf2vnlem4 14008. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vnlem2	⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))

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

Proof of Theorem bj-inf2vnlem2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2177	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2		eqeq1 2177	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = suc 𝑦 ↔ 𝑢 = suc 𝑦))
3	2	rexbidv 2471	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦))
4	1, 3	orbi12d 788	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦)))
5	4	rspcv 2830	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦)))
6		df-bj-ind 13962	. . . . . . . . 9 ⊢ (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣 ∈ 𝑍 suc 𝑣 ∈ 𝑍))
7	6	simplbi 272	. . . . . . . 8 ⊢ (Ind 𝑍 → ∅ ∈ 𝑍)
8		eleq1 2233	. . . . . . . 8 ⊢ (𝑢 = ∅ → (𝑢 ∈ 𝑍 ↔ ∅ ∈ 𝑍))
9	7, 8	syl5ibr 155	. . . . . . 7 ⊢ (𝑢 = ∅ → (Ind 𝑍 → 𝑢 ∈ 𝑍))
10	9	a1dd 48	. . . . . 6 ⊢ (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))
11		vex 2733	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
12	11	sucid 4402	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
13		eleq2 2234	. . . . . . . . . 10 ⊢ (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦 ↔ 𝑦 ∈ 𝑢))
14	13	eqcoms 2173	. . . . . . . . 9 ⊢ (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦 ↔ 𝑦 ∈ 𝑢))
15	12, 14	mpbii 147	. . . . . . . 8 ⊢ (𝑢 = suc 𝑦 → 𝑦 ∈ 𝑢)
16		eleq1 2233	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		eleq1 2233	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑍 ↔ 𝑦 ∈ 𝑍))
18	16, 17	imbi12d 233	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑦 → ((𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑍)))
19	18	rspcv 2830	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑍)))
20		bj-indsuc 13963	. . . . . . . . . . . 12 ⊢ (Ind 𝑍 → (𝑦 ∈ 𝑍 → suc 𝑦 ∈ 𝑍))
21		eleq1a 2242	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ 𝑍 → (𝑢 = suc 𝑦 → 𝑢 ∈ 𝑍))
22	20, 21	syl6com 35	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑍 → (Ind 𝑍 → (𝑢 = suc 𝑦 → 𝑢 ∈ 𝑍)))
23	19, 22	syl8 71	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑢 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑦 ∈ 𝐴 → (Ind 𝑍 → (𝑢 = suc 𝑦 → 𝑢 ∈ 𝑍)))))
24	23	com13 80	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑦 ∈ 𝑢 → (Ind 𝑍 → (𝑢 = suc 𝑦 → 𝑢 ∈ 𝑍)))))
25	24	com25 91	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (𝑢 = suc 𝑦 → (𝑦 ∈ 𝑢 → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))))
26	15, 25	mpdi 43	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍))))
27	26	rexlimiv 2581	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))
28	10, 27	jaoi 711	. . . . 5 ⊢ ((𝑢 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦) → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))
29	5, 28	syl6 33	. . . 4 ⊢ (𝑢 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍))))
30	29	com3l 81	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍))))
31	30	alrimdv 1869	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍))))
32		bi2.04 247	. . 3 ⊢ ((𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)) ↔ (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
33	32	albii 1463	. 2 ⊢ (∀𝑢(𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)) ↔ ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
34	31, 33	syl6ib 160	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703  ∀wal 1346   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  ∅c0 3414  suc csuc 4350  Ind wind 13961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-suc 4356  df-bj-ind 13962

This theorem is referenced by:  bj-inf2vnlem3  14007  bj-inf2vnlem4  14008