Theorem bj-inf2vnlem1 10095

Description: Lemma for bj-inf2vn 10099. 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 121	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴))
2		ax-io 630	. . . . . 6 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
3	2	biimpi 113	. . . . 5 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) → ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
4		simpl 102	. . . . . 6 ⊢ (((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)) → (𝑥 = ∅ → 𝑥 ∈ 𝐴))
5		eleq1 2100	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
6	4, 5	mpbidi 140	. . . . 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 1344	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9		exim 1490	. . 3 ⊢ (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10		0ex 3884	. . . . . 6 ⊢ ∅ ∈ V
11	10	isseti 2563	. . . . 5 ⊢ ∃𝑥 𝑥 = ∅
12		pm2.27 35	. . . . 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 9902	. . . 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 107	. . . . . 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 1344	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
20		eqid 2040	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
21		suceq 4139	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
22	21	eqeq2d 2051	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
23	22	rspcev 2656	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
24	20, 23	mpan2 401	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
25		vex 2560	. . . . . 6 ⊢ 𝑧 ∈ V
26	25	bj-sucex 10043	. . . . 5 ⊢ suc 𝑧 ∈ V
27		eqeq1 2046	. . . . . . 7 ⊢ (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
28	27	rexbidv 2327	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦))
29		eleq1 2100	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ 𝐴 ↔ suc 𝑧 ∈ 𝐴))
30	28, 29	imbi12d 223	. . . . 5 ⊢ (𝑥 = suc 𝑧 → ((∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴) ↔ (∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧 ∈ 𝐴)))
31	26, 30	spcv 2646	. . . 4 ⊢ (∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧 ∈ 𝐴))
32	19, 24, 31	syl2im 34	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → (𝑧 ∈ 𝐴 → suc 𝑧 ∈ 𝐴))
33	32	ralrimiv 2391	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴)
34		df-bj-ind 10051	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴))
35	16, 33, 34	sylanbrc 394	1 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  ∅c0 3224  suc csuc 4102  Ind wind 10050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-suc 4108  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bj-inf2vn  10099  bj-inf2vn2  10100