Theorem bj-inex 15553

Description: The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Proof of Theorem bj-inex

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

Step	Hyp	Ref	Expression
1		elisset 2777	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2777	. 2 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3		ax-17 1540	. . . 4 ⊢ (∃𝑦 𝑦 = 𝐵 → ∀𝑥∃𝑦 𝑦 = 𝐵)
4		19.29r 1635	. . . 4 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylan2 286	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6		ax-17 1540	. . . . 5 ⊢ (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7		19.29 1634	. . . . 5 ⊢ ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	6, 7	sylan 283	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9	8	eximi 1614	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		ineq12 3359	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
11	10	2eximi 1615	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		dfin5 3164	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
13		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
14		ax-bdel 15467	. . . . . . . . 9 ⊢ BOUNDED 𝑧 ∈ 𝑦
15		bdcv 15494	. . . . . . . . 9 ⊢ BOUNDED 𝑥
16	14, 15	bdrabexg 15552	. . . . . . . 8 ⊢ (𝑥 ∈ V → {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V)
17	13, 16	ax-mp 5	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V
18	12, 17	eqeltri 2269	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V
19		eleq1 2259	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → ((𝑥 ∩ 𝑦) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
20	18, 19	mpbii 148	. . . . 5 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
21	20	exlimivv 1911	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
22	11, 21	syl 14	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
23	5, 9, 22	3syl 17	. 2 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
24	1, 2, 23	syl2an 289	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {crab 2479  Vcvv 2763   ∩ cin 3156

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 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-ext 2178  ax-bd0 15459  ax-bdan 15461  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-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-bdc 15487

This theorem is referenced by:  speano5  15590