Theorem bj-inex 13276

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 2703	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2703	. 2 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3		ax-17 1507	. . . 4 ⊢ (∃𝑦 𝑦 = 𝐵 → ∀𝑥∃𝑦 𝑦 = 𝐵)
4		19.29r 1601	. . . 4 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylan2 284	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6		ax-17 1507	. . . . 5 ⊢ (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7		19.29 1600	. . . . 5 ⊢ ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	6, 7	sylan 281	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9	8	eximi 1580	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		ineq12 3277	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
11	10	2eximi 1581	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		dfin5 3083	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
13		vex 2692	. . . . . . . 8 ⊢ 𝑥 ∈ V
14		ax-bdel 13190	. . . . . . . . 9 ⊢ BOUNDED 𝑧 ∈ 𝑦
15		bdcv 13217	. . . . . . . . 9 ⊢ BOUNDED 𝑥
16	14, 15	bdrabexg 13275	. . . . . . . 8 ⊢ (𝑥 ∈ V → {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V)
17	13, 16	ax-mp 5	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} ∈ V
18	12, 17	eqeltri 2213	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ∈ V
19		eleq1 2203	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → ((𝑥 ∩ 𝑦) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
20	18, 19	mpbii 147	. . . . 5 ⊢ ((𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ V)
21	20	exlimivv 1869	. . . 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 287	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {crab 2421  Vcvv 2689   ∩ cin 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdan 13184  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089  df-bdc 13210

This theorem is referenced by:  speano5  13313