Theorem bj-inf2vnlem1 15462

Description: Lemma for bj-inf2vn 15466. 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 711	. . . . . 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 2256	. . . . . 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 1466	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9		exim 1610	. . 3 ⊢ (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10		0ex 4156	. . . . . 6 ⊢ ∅ ∈ V
11	10	isseti 2768	. . . . 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 15254	. . . 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 1466	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
20		eqid 2193	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
21		suceq 4433	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
22	21	eqeq2d 2205	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
23	22	rspcev 2864	. . . . 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 2763	. . . . . 6 ⊢ 𝑧 ∈ V
26	25	bj-sucex 15415	. . . . 5 ⊢ suc 𝑧 ∈ V
27		eqeq1 2200	. . . . . . 7 ⊢ (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
28	27	rexbidv 2495	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦))
29		eleq1 2256	. . . . . 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 2854	. . . 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 2566	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴)
34		df-bj-ind 15419	. 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 709  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∅c0 3446  suc csuc 4396  Ind wind 15418

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-suc 4402  df-bdc 15333  df-bj-ind 15419

This theorem is referenced by:  bj-inf2vn  15466  bj-inf2vn2  15467