Theorem elrabi 2841

Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Assertion

Ref	Expression
elrabi	⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrabi

Step	Hyp	Ref	Expression
1		clelab 2266	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)))
2		eleq1 2203	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	anbi1d 461	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ (𝐴 ∈ 𝑉 ∧ 𝜑)))
4	3	simprbda 381	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
5	4	exlimiv 1578	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
6	1, 5	sylbi 120	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} → 𝐴 ∈ 𝑉)
7		df-rab 2426	. 2 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
8	6, 7	eleq2s 2235	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  {crab 2421

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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426

This theorem is referenced by:  ordtriexmidlem  4443  ordtri2or2exmidlem  4449  onsucelsucexmidlem  4452  ordsoexmid  4485  reg3exmidlemwe  4501  elfvmptrab1  5523  acexmidlemcase  5777  ssfirab  6830  exmidonfinlem  7066  cc4f  7101  genpelvl  7344  genpelvu  7345  suplocsrlempr  7639  nnindnn  7725  sup3exmid  8739  nnind  8760  supinfneg  9417  infsupneg  9418  supminfex  9419  ublbneg  9432  hashinfuni  10555  zsupcllemstep  11674  infssuzex  11678  infssuzledc  11679  bezoutlemsup  11733  lcmgcdlem  11794  oddennn  11941  evenennn  11942  znnen  11947  ennnfonelemg  11952  txdis1cn  12486  reopnap  12746  divcnap  12763  limccl  12836  dvlemap  12857  dvaddxxbr  12873  dvmulxxbr  12874  dvcoapbr  12879  dvcjbr  12880  dvrecap  12885  dveflem  12895