Theorem elin1d 3339

Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)

Hypothesis

Ref	Expression
elin1d.1	|- ( ph -> X e. ( A i^i B ) )

Assertion

Ref	Expression
elin1d	|- ( ph -> X e. A )

Proof of Theorem elin1d

Step	Hyp	Ref	Expression
1		elin1d.1	. 2 |- ( ph -> X e. ( A i^i B ) )
2		elinel1 3336	. 2 |- ( X e. ( A i^i B ) -> X e. A )
3	1, 2	syl 14	1 |- ( ph -> X e. A )

Syntax hints:   -> wi 4   e. wcel 2160   i^i cin 3143

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  fiuni  7008  explecnv  11548  nninfdclemcl  12502  nninfdclemp1  12504  2idllidld  13838  qus1  13858  restbasg  14145  txcnp  14248  blin2  14409  bj-charfun  15037