Theorem bj-indint 7154

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 7150	. . . . 5 ⊢ (Ind x ↔ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x))
2	1	simplbi 259	. . . 4 ⊢ (Ind x → ∅ ∈ x)
3	2	rgenw 2354	. . 3 ⊢ ∀x ∈ A (Ind x → ∅ ∈ x)
4		0ex 3858	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3601	. . 3 ⊢ (∅ ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → ∅ ∈ x))
6	3, 5	mpbir 134	. 2 ⊢ ∅ ∈ ∩ {x ∈ A ∣ Ind x}
7		bj-indsuc 7151	. . . . . 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 2362	. . . 4 ⊢ (∀x ∈ A (Ind x → y ∈ x) → ∀x ∈ A (Ind x → suc y ∈ x))
10		vex 2538	. . . . 5 ⊢ y ∈ V
11	10	elintrab 3601	. . . 4 ⊢ (y ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → y ∈ x))
12	10	bj-sucex 7146	. . . . 5 ⊢ suc y ∈ V
13	12	elintrab 3601	. . . 4 ⊢ (suc y ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → suc y ∈ x))
14	9, 11, 13	3imtr4i 190	. . 3 ⊢ (y ∈ ∩ {x ∈ A ∣ Ind x} → suc y ∈ ∩ {x ∈ A ∣ Ind x})
15	14	rgen 2352	. 2 ⊢ ∀y ∈ ∩ {x ∈ A ∣ Ind x}suc y ∈ ∩ {x ∈ A ∣ Ind x}
16		df-bj-ind 7150	. 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 837	1 ⊢ Ind ∩ {x ∈ A ∣ Ind x}

Syntax hints:   → wi 4   ∈ wcel 1374  ∀wral 2284  {crab 2288  ∅c0 3201  ∩ cint 3589  suc csuc 4051  Ind wind 7149

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-bdc 7068  df-bj-ind 7150

This theorem is referenced by:  bj-omind  7156