MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brtpos0 Unicode version

Theorem brtpos0 6981
Description: The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on A, in brtpos 6983. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 6980 . 2
2 ssun2 3667 . . . . 5
3 0ex 4582 . . . . . 6
43snid 4057 . . . . 5
52, 4sselii 3500 . . . 4
65biantrur 506 . . 3
7 cnvsn0 5481 . . . . . 6
87unieqi 4258 . . . . 5
9 uni0 4276 . . . . 5
108, 9eqtri 2486 . . . 4
1110breq1i 4459 . . 3
126, 11bitr3i 251 . 2
131, 12syl6bb 261 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  u.cun 3473   c0 3784  {csn 4029  U.cuni 4249   class class class wbr 4452  `'ccnv 5003  domcdm 5004  tposctpos 6973
This theorem is referenced by:  reldmtpos  6982  brtpos  6983  tpostpos  6994
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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592
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-eu 2286  df-mo 2287  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-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-tpos 6974
  Copyright terms: Public domain W3C validator