Theorem brrelex12 5042

Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex12

Proof of Theorem brrelex12

Step	Hyp	Ref	Expression
1		df-rel 5011	. . . . 5
2	1	biimpi 194	. . . 4
3	2	ssbrd 4493	. . 3
4	3	imp 429	. 2
5		brxp 5035	. 2
6	4, 5	sylib 196	1

Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  cvv 3109  C_wss 3475   class class class wbr 4452  X.cxp 5002  Relwrel 5009

This theorem is referenced by:  brrelex  5043  brrelex2  5044  relbrcnvg  5380  ovprc  6326  oprabv  6345  brovex  6969  ersym  7342  relelec  7371  encv  7544  fsuppunbi  7870  fpwwe2lem2  9031  fpwwelem  9044  isstruct2  14641  brssc  15183  cofuval2  15256  isfull  15279  isfth  15283  isnat  15316  pslem  15836  frgpuplem  16790  dvdsr  17295  tpsexOLD  19420  ulmval  22775  perpln1  24087  perpln2  24088  iseupa  24965  rngoablo2  25424  opelco3  29208  aovprc  32273  aovrcl  32274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011