Theorem elrabi 2933

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 2333	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)))
2		eleq1 2270	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	anbi1d 465	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ (𝐴 ∈ 𝑉 ∧ 𝜑)))
4	3	simprbda 383	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
5	4	exlimiv 1622	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉)
6	1, 5	sylbi 121	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} → 𝐴 ∈ 𝑉)
7		df-rab 2495	. 2 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
8	6, 7	eleq2s 2302	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2178  {cab 2193  {crab 2490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-rab 2495

This theorem is referenced by:  ordtriexmidlem  4585  ordtri2or2exmidlem  4592  onsucelsucexmidlem  4595  ordsoexmid  4628  reg3exmidlemwe  4645  elfvmptrab1  5697  acexmidlemcase  5962  elovmporab  6169  elovmporab1w  6170  ssfirab  7059  exmidonfinlem  7332  cc4f  7416  genpelvl  7660  genpelvu  7661  suplocsrlempr  7955  nnindnn  8041  sup3exmid  9065  nnind  9087  supinfneg  9751  infsupneg  9752  supminfex  9753  ublbneg  9769  zsupcllemstep  10409  infssuzex  10413  infssuzledc  10414  hashinfuni  10959  bezoutlemsup  12445  uzwodc  12473  nninfctlemfo  12476  lcmgcdlem  12514  phisum  12678  oddennn  12878  evenennn  12879  znnen  12884  ennnfonelemg  12889  rrgval  14139  psrbagf  14547  txdis1cn  14865  reopnap  15133  divcnap  15152  limccl  15246  dvlemap  15267  dvaddxxbr  15288  dvmulxxbr  15289  dvcoapbr  15294  dvcjbr  15295  dvrecap  15300  dveflem  15313  sgmval  15570  0sgm  15572  sgmf  15573  sgmnncl  15575  dvdsppwf1o  15576  sgmppw  15579  uhgrss  15786