Theorem bdinex1 16669

Description: Bounded version of inex1 4244. (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 16616	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵
4	3	bdzfauscl 16660	. . . 4 ⊢ (𝐴 ∈ V → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
5	1, 4	ax-mp 5	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6		dfcleq 2226	. . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)))
7		elin 3402	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	7	bibi2i 227	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	8	albii 1519	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
10	6, 9	bitri 184	. . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
11	10	exbii 1654	. . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
12	5, 11	mpbir 146	. 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵)
13	12	issetri 2823	1 ⊢ (𝐴 ∩ 𝐵) ∈ V

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813   ∩ cin 3210  BOUNDED wbdc 16610

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bdsep 16654

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-bdc 16611

This theorem is referenced by:  bdinex2  16670  bdinex1g  16671  bdpeano5  16713