Theorem bdinex1 11447

Description: Bounded version of inex1 3965. (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 11394	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵
4	3	bdzfauscl 11438	. . . 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 2082	. . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)))
7		elin 3181	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	7	bibi2i 225	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	8	albii 1404	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
10	6, 9	bitri 182	. . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
11	10	exbii 1541	. . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
12	5, 11	mpbir 144	. 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵)
13	12	issetri 2628	1 ⊢ (𝐴 ∩ 𝐵) ∈ V

Syntax hints:   ∧ wa 102   ↔ wb 103  ∀wal 1287   = wceq 1289  ∃wex 1426   ∈ wcel 1438  Vcvv 2619   ∩ cin 2996  BOUNDED wbdc 11388

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bdsep 11432

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-bdc 11389

This theorem is referenced by:  bdinex2  11448  bdinex1g  11449  bdpeano5  11495