Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trintALTVD Structured version   Visualization version   Unicode version

Theorem trintALTVD 37277
Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 37278. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 37278 is trintALTVD 37277 without virtual deductions and was automatically derived from trintALTVD 37277.
 1:: 2:: 3:2: 4:2: 5:4: 6:5: 7:: 8:7,6: 9:7,1: 10:7,9: 11:10,3,8: 12:11: 13:12: 14:13: 15:3,14: 16:15: 17:16: 18:17: qed:18:
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintALTVD
Distinct variable group:   ,

Proof of Theorem trintALTVD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn2 36992 . . . . . . 7
2 simpl 459 . . . . . . 7
31, 2e2 37010 . . . . . 6
4 idn3 36994 . . . . . . . . . . 11
5 idn1 36944 . . . . . . . . . . . 12
6 rspsbc 3346 . . . . . . . . . . . 12
74, 5, 6e31 37138 . . . . . . . . . . 11
8 trsbc 36901 . . . . . . . . . . . 12
98biimpd 211 . . . . . . . . . . 11
104, 7, 9e33 37121 . . . . . . . . . 10
11 simpr 463 . . . . . . . . . . . . . 14
121, 11e2 37010 . . . . . . . . . . . . 13
13 elintg 4242 . . . . . . . . . . . . . 14
1413ibi 245 . . . . . . . . . . . . 13
1512, 14e2 37010 . . . . . . . . . . . 12
16 rsp 2754 . . . . . . . . . . . 12
1715, 16e2 37010 . . . . . . . . . . 11
18 pm2.27 40 . . . . . . . . . . 11
194, 17, 18e32 37145 . . . . . . . . . 10
20 trel 4504 . . . . . . . . . . 11
2120expd 438 . . . . . . . . . 10
2210, 3, 19, 21e323 37153 . . . . . . . . 9
2322in3 36988 . . . . . . . 8
2423gen21 36998 . . . . . . 7
25 df-ral 2742 . . . . . . . 8
2625biimpri 210 . . . . . . 7
2724, 26e2 37010 . . . . . 6
28 elintg 4242 . . . . . . 7
2928biimprd 227 . . . . . 6
303, 27, 29e22 37050 . . . . 5
3130in2 36984 . . . 4
3231gen12 36997 . . 3
33 dftr2 4499 . . . 4
3433biimpri 210 . . 3
3532, 34e1a 37006 . 2
3635in1 36941 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371  wal 1442   wcel 1887  wral 2737  wsbc 3267  cint 4234   wtr 4497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-v 3047  df-sbc 3268  df-in 3411  df-ss 3418  df-uni 4199  df-int 4235  df-tr 4498  df-vd1 36940  df-vd2 36948  df-vd3 36960 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator