Theorem bdinex1 13268

Description: Bounded version of inex1 4070. (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 13215	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵
4	3	bdzfauscl 13259	. . . 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 2134	. . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)))
7		elin 3264	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	7	bibi2i 226	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	8	albii 1447	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
10	6, 9	bitri 183	. . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
11	10	exbii 1585	. . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
12	5, 11	mpbir 145	. 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵)
13	12	issetri 2698	1 ⊢ (𝐴 ∩ 𝐵) ∈ V

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689   ∩ cin 3075  BOUNDED wbdc 13209

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-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-v 2691  df-in 3082  df-bdc 13210

This theorem is referenced by:  bdinex2  13269  bdinex1g  13270  bdpeano5  13312