Theorem elin1d 3306

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 3303	. 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 2135   i^i cin 3110

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-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117

This theorem is referenced by:  fiuni  6934  explecnv  11432  nninfdclemcl  12320  nninfdclemp1  12322  restbasg  12709  txcnp  12812  blin2  12973  bj-charfun  13524