Theorem inex1 4269

Description: Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
inex1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
inex1	⊢ (𝐴 ∩ 𝐵) ∈ V

Proof of Theorem inex1

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

Step	Hyp	Ref	Expression
1		inex1.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	sepgi 4254	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		dfcleq 2232	. . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)))
4		elin 3412	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	bibi2i 227	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
6	5	albii 1523	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	3, 6	bitri 184	. . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
8	7	exbii 1658	. . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	2, 8	mpbir 146	. 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵)
10	9	issetri 2831	1 ⊢ (𝐴 ∩ 𝐵) ∈ V

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4251

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  inex2  4270  inex1g  4271  inuni  4293  bnd2  4312  peano5  4747  ssimaex  5767  ofmres  6373  tfrexlem  6609  elrest  13659  epttop  15316  tgrest  15395  resttopon  15397  restco  15400  cnrest2  15462  cnptopresti  15464  cnptoprest  15465  cnptoprest2  15466  txrest  15502