Theorem elrabi 2695

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 2162	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)))
2		eleq1 2100	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	anbi1d 438	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ (𝐴 ∈ 𝑉 ∧ 𝜑)))
4	3	simprbda 365	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
5	4	exlimiv 1489	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
6	1, 5	sylbi 114	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} → 𝐴 ∈ 𝑉)
7		df-rab 2315	. 2 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
8	6, 7	eleq2s 2132	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  {crab 2310

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rab 2315

This theorem is referenced by:  ordtriexmidlem  4245  ordtri2or2exmidlem  4251  onsucelsucexmidlem  4254  ordsoexmid  4286  reg3exmidlemwe  4303  acexmidlemcase  5507  genpelvl  6610  genpelvu  6611  nnindnn  6967  nnind  7930  ublbneg  8548