Theorem bropaex12 5192

Description: Two classes related by an ordered pair class builder are sets. (Contributed by AV, 21-Jan-2020.)

Hypothesis

Ref	Expression
bropaex12.1	|- R = { <. x , y >. | ps }

Assertion

Ref	Expression
bropaex12	|- ( A R B -> ( A e. _V /\ B e. _V ) )

Distinct variable groups:   x, A, y   x, B, y

Allowed substitution hints:   ps( x, y)   R( x, y)

Proof of Theorem bropaex12

Step	Hyp	Ref	Expression
1		df-br 4654	. . . 4 |- ( A R B <-> <. A , B >. e. R )
2		bropaex12.1	. . . . 5 |- R = { <. x , y >. | ps }
3	2	eleq2i 2693	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ps } )
4	1, 3	bitri 264	. . 3 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ps } )
5		elopaelxp 5191	. . 3 |- ( <. A , B >. e. { <. x , y >. | ps } -> <. A , B >. e. ( _V X. _V ) )
6	4, 5	sylbi 207	. 2 |- ( A R B -> <. A , B >. e. ( _V X. _V ) )
7		opelxp 5146	. 2 |- ( <. A , B >. e. ( _V X. _V ) <-> ( A e. _V /\ B e. _V ) )
8	6, 7	sylib 208	1 |- ( A R B -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {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-ral 2917  df-rex 2918  df-rab 2921  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-br 4654  df-opab 4713  df-xp 5120

This theorem is referenced by:  fpwwe  9468  efgrelexlema  18162  brsslt  31900  clcllaw  41827  asslawass  41829