Theorem bj-inf2vnlem1 12138

Description: Lemma for bj-inf2vn 12142. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vnlem1	⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem bj-inf2vnlem1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bi2 129	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴))
2		jaob 667	. . . . . 6 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
3	2	biimpi 119	. . . . 5 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) → ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
4		simpl 108	. . . . . 6 ⊢ (((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)) → (𝑥 = ∅ → 𝑥 ∈ 𝐴))
5		eleq1 2151	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
6	4, 5	mpbidi 150	. . . . 5 ⊢ (((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
7	1, 3, 6	3syl 17	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
8	7	alimi 1390	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9		exim 1536	. . 3 ⊢ (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10		0ex 3972	. . . . . 6 ⊢ ∅ ∈ V
11	10	isseti 2628	. . . . 5 ⊢ ∃𝑥 𝑥 = ∅
12		pm2.27 40	. . . . 5 ⊢ (∃𝑥 𝑥 = ∅ → ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴))
13	11, 12	ax-mp 7	. . . 4 ⊢ ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴)
14		bj-ex 11936	. . . 4 ⊢ (∃𝑥∅ ∈ 𝐴 → ∅ ∈ 𝐴)
15	13, 14	syl 14	. . 3 ⊢ ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∅ ∈ 𝐴)
16	8, 9, 15	3syl 17	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∅ ∈ 𝐴)
17	3	simprd 113	. . . . . 6 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
18	1, 17	syl 14	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
19	18	alimi 1390	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
20		eqid 2089	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
21		suceq 4238	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
22	21	eqeq2d 2100	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
23	22	rspcev 2723	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
24	20, 23	mpan2 417	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
25		vex 2623	. . . . . 6 ⊢ 𝑧 ∈ V
26	25	bj-sucex 12087	. . . . 5 ⊢ suc 𝑧 ∈ V
27		eqeq1 2095	. . . . . . 7 ⊢ (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
28	27	rexbidv 2382	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦))
29		eleq1 2151	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ 𝐴 ↔ suc 𝑧 ∈ 𝐴))
30	28, 29	imbi12d 233	. . . . 5 ⊢ (𝑥 = suc 𝑧 → ((∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴) ↔ (∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧 ∈ 𝐴)))
31	26, 30	spcv 2713	. . . 4 ⊢ (∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧 ∈ 𝐴))
32	19, 24, 31	syl2im 38	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (𝑧 ∈ 𝐴 → suc 𝑧 ∈ 𝐴))
33	32	ralrimiv 2446	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴)
34		df-bj-ind 12095	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴))
35	16, 33, 34	sylanbrc 409	1 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 665  ∀wal 1288   = wceq 1290  ∃wex 1427   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361  ∅c0 3287  suc csuc 4201  Ind wind 12094

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-in1 580  ax-in2 581  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-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-nul 3971  ax-pr 4045  ax-un 4269  ax-bd0 11977  ax-bdor 11980  ax-bdex 11983  ax-bdeq 11984  ax-bdel 11985  ax-bdsb 11986  ax-bdsep 12048

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-dif 3002  df-un 3004  df-nul 3288  df-sn 3456  df-pr 3457  df-uni 3660  df-suc 4207  df-bdc 12005  df-bj-ind 12095

This theorem is referenced by:  bj-inf2vn  12142  bj-inf2vn2  12143