Theorem elex2 2742

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Assertion

Ref	Expression
elex2	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2

Step	Hyp	Ref	Expression
1		eleq1a 2238	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
2	1	alrimiv 1862	. 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3		elisset 2740	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
4		exim 1587	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵))
5	2, 3, 4	sylc 62	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  snmg  3694  oprcl  3782  brm  4032  ss1o0el1  4176  exss  4205  onintrab2im  4495  regexmidlemm  4509  dmxpid  4825  acexmidlem2  5839  frecabcl  6367  ixpm  6696  enm  6786  ssfilem  6841  fin0  6851  fin0or  6852  diffitest  6853  diffisn  6859  infm  6870  inffiexmid  6872  ctssdc  7078  omct  7082  ctssexmid  7114  exmidfodomrlemr  7158  exmidfodomrlemrALT  7159  exmidaclem  7164  caucvgsrlemasr  7731  suplocsrlempr  7748  gtso  7977  sup3exmid  8852  indstr  9531  negm  9553  fzm  9973  fzom  10099  rexfiuz  10931  r19.2uz  10935  resqrexlemgt0  10962  climuni  11234  bezoutlembi  11938  lcmgcdlem  12009  pcprecl  12221  pc2dvds  12261  nninfdclemcl  12381  tgioo  13186  pw1nct  13883  nninfall  13889