Theorem elopaelxp 5191

Description: Membership in an ordered pair class builder implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.)

Assertion

Ref	Expression
elopaelxp	|- ( A e. { <. x , y >. | ps } -> A e. ( _V X. _V ) )

Distinct variable group:   x, A, y

Allowed substitution hints:   ps( x, y)

Proof of Theorem elopaelxp

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( A = <. x , y >. /\ ps ) -> A = <. x , y >. )
2	1	2eximi 1763	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ps ) -> E. x E. y A = <. x , y >. )
3		elopab 4983	. 2 |- ( A e. { <. x , y >. | ps } <-> E. x E. y ( A = <. x , y >. /\ ps ) )
4		elvv 5177	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
5	2, 3, 4	3imtr4i 281	1 |- ( A e. { <. x , y >. | ps } -> A e. ( _V X. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  bropaex12  5192  clwlkcompim  26676  linedegen  32250  opelopab3  33511