Theorem bj-indint 15577

Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indint	⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-indint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 15573	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
2	1	simplbi 274	. . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥)
3	2	rgenw 2552	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4		0ex 4160	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3886	. . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
6	3, 5	mpbir 146	. 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
7		bj-indsuc 15574	. . . . . 6 ⊢ (Ind 𝑥 → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
8	7	a2i 11	. . . . 5 ⊢ ((Ind 𝑥 → 𝑦 ∈ 𝑥) → (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
9	8	ralimi 2560	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
10		vex 2766	. . . . 5 ⊢ 𝑦 ∈ V
11	10	elintrab 3886	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥))
12	10	bj-sucex 15569	. . . . 5 ⊢ suc 𝑦 ∈ V
13	12	elintrab 3886	. . . 4 ⊢ (suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
14	9, 11, 13	3imtr4i 201	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} → suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥})
15	14	rgen 2550	. 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
16		df-bj-ind 15573	. 2 ⊢ (Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}))
17	6, 15, 16	mpbir2an 944	1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Syntax hints:   → wi 4   ∈ wcel 2167  ∀wral 2475  {crab 2479  ∅c0 3450  ∩ cint 3874  suc csuc 4400  Ind wind 15572

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4159  ax-pr 4242  ax-un 4468  ax-bd0 15459  ax-bdor 15462  ax-bdex 15465  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-nul 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-bdc 15487  df-bj-ind 15573

This theorem is referenced by:  bj-omind  15580