Theorem bj-inf2vnlem1 14725

Description: Lemma for bj-inf2vn 14729. 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 710	. . . . . 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 2240	. . . . . 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 1455	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9		exim 1599	. . 3 ⊢ (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10		0ex 4131	. . . . . 6 ⊢ ∅ ∈ V
11	10	isseti 2746	. . . . 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 14517	. . . 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 1455	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 → 𝑥 ∈ 𝐴))
20		eqid 2177	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
21		suceq 4403	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
22	21	eqeq2d 2189	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
23	22	rspcev 2842	. . . . 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 2741	. . . . . 6 ⊢ 𝑧 ∈ V
26	25	bj-sucex 14678	. . . . 5 ⊢ suc 𝑧 ∈ V
27		eqeq1 2184	. . . . . . 7 ⊢ (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
28	27	rexbidv 2478	. . . . . 6 ⊢ (𝑥 = suc 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦 ∈ 𝐴 suc 𝑧 = suc 𝑦))
29		eleq1 2240	. . . . . 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 2832	. . . 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 2549	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 suc 𝑧 ∈ 𝐴)
34		df-bj-ind 14682	. 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 708  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  ∅c0 3423  suc csuc 4366  Ind wind 14681

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4130  ax-pr 4210  ax-un 4434  ax-bd0 14568  ax-bdor 14571  ax-bdex 14574  ax-bdeq 14575  ax-bdel 14576  ax-bdsb 14577  ax-bdsep 14639

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-nul 3424  df-sn 3599  df-pr 3600  df-uni 3811  df-suc 4372  df-bdc 14596  df-bj-ind 14682

This theorem is referenced by:  bj-inf2vn  14729  bj-inf2vn2  14730