Theorem bj-inf2vnlem1 16565

Description: Lemma for bj-inf2vn 16569. 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		biimpr 130	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴))
2		jaob 717	. . . . . 6 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
3	2	biimpi 120	. . . . 5 ⊢ (((𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → 𝑥 ∈ 𝐴) → ((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)))
4		simpl 109	. . . . . 6 ⊢ (((𝑥 = ∅ → 𝑥 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴)) → (𝑥 = ∅ → 𝑥 ∈ 𝐴))
5		eleq1 2294	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
6	4, 5	mpbidi 151	. . . . 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 1503	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9		exim 1647	. . 3 ⊢ (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10		0ex 4216	. . . . . 6 ⊢ ∅ ∈ V
11	10	isseti 2811	. . . . 5 ⊢ ∃𝑥 𝑥 = ∅
12		pm2.27 40	. . . . 5 ⊢ (∃𝑥 𝑥 = ∅ → ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴))
13	11, 12	ax-mp 5	. . . 4 ⊢ ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴)
14		bj-ex 16358	. . . 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 114	. . . . . 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 1503	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
20		eqid 2231	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
21		suceq 4499	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
22	21	eqeq2d 2243	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
23	22	rspcev 2910	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
24	20, 23	mpan2 425	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦)
25		vex 2805	. . . . . 6 ⊢ 𝑧 ∈ V
26	25	bj-sucex 16518	. . . . 5 ⊢ suc 𝑧 ∈ V
27		eqeq1 2238	. . . . . . 7 ⊢ (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
28	27	rexbidv 2533	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦))
29		eleq1 2294	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ 𝐴 ↔ suc 𝑧 ∈ 𝐴))
30	28, 29	imbi12d 234	. . . . 5 ⊢ (𝑥 = suc 𝑧 → ((∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴) ↔ (∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧 ∈ 𝐴)))
31	26, 30	spcv 2900	. . . 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 2604	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴)
34		df-bj-ind 16522	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴))
35	16, 33, 34	sylanbrc 417	1 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  ∅c0 3494  suc csuc 4462  Ind wind 16521

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16408  ax-bdor 16411  ax-bdex 16414  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417  ax-bdsep 16479

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-suc 4468  df-bdc 16436  df-bj-ind 16522

This theorem is referenced by:  bj-inf2vn  16569  bj-inf2vn2  16570