Theorem bj-inex 10027

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 2568	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2568	. 2 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3		ax-17 1419	. . . 4 ⊢ (∃𝑦 𝑦 = 𝐵 → ∀𝑥∃𝑦 𝑦 = 𝐵)
4		19.29r 1512	. . . 4 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylan2 270	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6		ax-17 1419	. . . . 5 ⊢ (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7		19.29 1511	. . . . 5 ⊢ ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	6, 7	sylan 267	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9	8	eximi 1491	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		ineq12 3133	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
11	10	2eximi 1492	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		dfin5 2925	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
13		vex 2560	. . . . . . . 8 ⊢ 𝑥 ∈ V
14		ax-bdel 9941	. . . . . . . . 9 ⊢ BOUNDED 𝑧 ∈ 𝑦
15		bdcv 9968	. . . . . . . . 9 ⊢ BOUNDED 𝑥
16	14, 15	bdrabexg 10026	. . . . . . . 8 ⊢ (𝑥 ∈ V → {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V)
17	13, 16	ax-mp 7	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V
18	12, 17	eqeltri 2110	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V
19		eleq1 2100	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → ((𝑥 ∩ 𝑦) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
20	18, 19	mpbii 136	. . . . 5 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
21	20	exlimivv 1776	. . . 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 273	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {crab 2310  Vcvv 2557   ∩ cin 2916

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by:  speano5  10069