Theorem brrelex1i 4818

Description: The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)

Hypothesis

Ref	Expression
brrelexi.1	⊢ Rel 𝑅

Assertion

Ref	Expression
brrelex1i	⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)

Proof of Theorem brrelex1i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 ⊢ Rel 𝑅
2		brrelex1 4814	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
3	1, 2	mpan 428	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  Vcvv 2821   class class class wbr 4130  Rel wrel 4779

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-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781

This theorem is used by:  nprrel  4820  vtoclr  4823  opeliunxp2  4920  ideqg  4931  issetid  4934  fvmptss2  5780  opeliunxp2f  6509  brtpos2  6522  brdomg  7032  ctex  7037  isfi  7047  domssr  7064  en1uniel  7091  xpdom2  7129  xpdom1g  7131  xpen  7145  isbth  7284  relprcnfsupp  7288  djudom  7433  cc3  7634  aprcl  8974  climcl  12048  climi  12053  climrecl  12090  structex  13364