Theorem bdinex1 9354

Description: Bounded version of inex1 3882. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdinex1.bd	⊢ BOUNDED B
bdinex1.1	⊢ A ∈ V

Assertion

Ref	Expression
bdinex1	⊢ (A ∩ B) ∈ V

Proof of Theorem bdinex1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdinex1.1	. . . 4 ⊢ A ∈ V
2		bdinex1.bd	. . . . . 6 ⊢ BOUNDED B
3	2	bdeli 9301	. . . . 5 ⊢ BOUNDED y ∈ B
4	3	bdzfauscl 9345	. . . 4 ⊢ (A ∈ V → ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B)))
5	1, 4	ax-mp 7	. . 3 ⊢ ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B))
6		dfcleq 2031	. . . . 5 ⊢ (x = (A ∩ B) ↔ ∀y(y ∈ x ↔ y ∈ (A ∩ B)))
7		elin 3120	. . . . . . 7 ⊢ (y ∈ (A ∩ B) ↔ (y ∈ A ∧ y ∈ B))
8	7	bibi2i 216	. . . . . 6 ⊢ ((y ∈ x ↔ y ∈ (A ∩ B)) ↔ (y ∈ x ↔ (y ∈ A ∧ y ∈ B)))
9	8	albii 1356	. . . . 5 ⊢ (∀y(y ∈ x ↔ y ∈ (A ∩ B)) ↔ ∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B)))
10	6, 9	bitri 173	. . . 4 ⊢ (x = (A ∩ B) ↔ ∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B)))
11	10	exbii 1493	. . 3 ⊢ (∃x x = (A ∩ B) ↔ ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B)))
12	5, 11	mpbir 134	. 2 ⊢ ∃x x = (A ∩ B)
13	12	issetri 2558	1 ⊢ (A ∩ B) ∈ V

Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910  BOUNDED wbdc 9295

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-bdc 9296

This theorem is referenced by:  bdinex2  9355  bdinex1g  9356  bdpeano5  9403