Theorem bj-inf2vnlem2 16911

Description: Lemma for bj-inf2vnlem3 16912 and bj-inf2vnlem4 16913. 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 2245	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2		eqeq1 2245	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = suc 𝑦 ↔ 𝑢 = suc 𝑦))
3	2	rexbidv 2551	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦))
4	1, 3	orbi12d 805	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦)))
5	4	rspcv 2925	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦)))
6		df-bj-ind 16867	. . . . . . . . 9 ⊢ (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣 ∈ 𝑍 suc 𝑣 ∈ 𝑍))
7	6	simplbi 274	. . . . . . . 8 ⊢ (Ind 𝑍 → ∅ ∈ 𝑍)
8		eleq1 2301	. . . . . . . 8 ⊢ (𝑢 = ∅ → (𝑢 ∈ 𝑍 ↔ ∅ ∈ 𝑍))
9	7, 8	imbitrrid 156	. . . . . . 7 ⊢ (𝑢 = ∅ → (Ind 𝑍 → 𝑢 ∈ 𝑍))
10	9	a1dd 48	. . . . . 6 ⊢ (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))
11		vex 2824	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
12	11	sucid 4557	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
13		eleq2 2302	. . . . . . . . . 10 ⊢ (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦 ↔ 𝑦 ∈ 𝑢))
14	13	eqcoms 2241	. . . . . . . . 9 ⊢ (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦 ↔ 𝑦 ∈ 𝑢))
15	12, 14	mpbii 148	. . . . . . . 8 ⊢ (𝑢 = suc 𝑦 → 𝑦 ∈ 𝑢)
16		eleq1 2301	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		eleq1 2301	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑍 ↔ 𝑦 ∈ 𝑍))
18	16, 17	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑦 → ((𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑍)))
19	18	rspcv 2925	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑍)))
20		bj-indsuc 16868	. . . . . . . . . . . 12 ⊢ (Ind 𝑍 → (𝑦 ∈ 𝑍 → suc 𝑦 ∈ 𝑍))
21		eleq1a 2310	. . . . . . . . . . . 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 2662	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)))
28	10, 27	jaoi 728	. . . . 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 1929	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍))))
32		bi2.04 248	. . 3 ⊢ ((𝑢 ∈ 𝐴 → (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → 𝑢 ∈ 𝑍)) ↔ (∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))
33	32	albii 1523	. 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	imbitrdi 161	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 105   ∨ wo 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∅c0 3520  suc csuc 4505  Ind wind 16866

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 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

This theorem 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-sn 3711  df-suc 4511  df-bj-ind 16867

This theorem is referenced by:  bj-inf2vnlem3  16912  bj-inf2vnlem4  16913