Type  Label  Description 
Statement 

Theorem  unipwrVD 37201 
Virtual deduction proof of unipwr 37202. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  unipwr 37202 
A class is a subclass of the union of its power class. This theorem is
the righttoleft subclass lemma of unipw 4671. The proof of this theorem
was automatically generated from unipwrVD 37201 using a tools command file ,
translateMWO.cmd , by translating the proof into its nonvirtual
deduction form and minimizing it. (Contributed by Alan Sare,
25Aug2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  sstrALT2VD 37203 
Virtual deduction proof of sstrALT2 37204. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  sstrALT2 37204 
Virtual deduction proof of sstr 3472, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 37203 using the command file translate_{without}_overwriting.cmd .
It was not minimized because the automated minimization excluding
duplicates generates a minimized proof which, although not directly
containing any duplicates, indirectly contains a duplicate. That is,
the trace back of the minimized proof contains a duplicate. This is
undesirable because some step(s) of the minimized proof use the proven
theorem. (Contributed by Alan Sare, 11Sep2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  suctrALT2VD 37205 
Virtual deduction proof of suctrALT2 37206. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  suctrALT2 37206 
Virtual deduction proof of suctr 5525. The sucessor of a transitive class
is transitive. This proof was generated automatically from the virtual
deduction proof suctrALT2VD 37205 using the tools command file
translate_{without}_overwriting_{minimize}_excluding_{duplicates}.cmd .
(Contributed by Alan Sare, 11Sep2011.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  elex2VD 37207* 
Virtual deduction proof of elex2 3092. (Contributed by Alan Sare,
25Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  elex22VD 37208* 
Virtual deduction proof of elex22 3093. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  eqsbc3rVD 37209* 
Virtual deduction proof of eqsbc3r 3356. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  zfregs2VD 37210* 
Virtual deduction proof of zfregs2 8225. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  tpid3gVD 37211 
Virtual deduction proof of tpid3g 4115. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem1VD 37212* 
Virtual deduction proof of en3lplem1 8128. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem2VD 37213* 
Virtual deduction proof of en3lplem2 8129. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lpVD 37214 
Virtual deduction proof of en3lp 8130. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



21.29.7 Theorems proved using Virtual Deduction
with mmj2 assistance


Theorem  simplbi2VD 37215 
Virtual deduction proof of simplbi2 629. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
h1:: 
 3:1,?: e0a 37132 
 qed:3,?: e0a 37132 

The proof of simplbi2 629 was automatically derived from it.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3ornot23VD 37216 
Virtual deduction proof of 3ornot23 36835. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1::
 2:: 
 3:1,?: e1a 36977 
 4:1,?: e1a 36977 
 5:3,4,?: e11 37038 
 6:2,?: e2 36981 
 7:5,6,?: e12 37084 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  orbi1rVD 37217 
Virtual deduction proof of orbi1r 36836. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:2,?: e2 36981 
 4:1,3,?: e12 37084 
 5:4,?: e2 36981 
 6:5: 
 7:: 
 8:7,?: e2 36981 
 9:1,8,?: e12 37084 
 10:9,?: e2 36981 
 11:10: 
 12:6,11,?: e11 37038 
 qed:12: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  bitr3VD 37218 
Virtual deduction proof of bitr3 36837. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:: 
 4:3,?: e2 36981 
 5:2,4,?: e12 37084 
 6:5: 
 qed:6: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3orbi123VD 37219 
Virtual deduction proof of 3orbi123 36838. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:1,?: e1a 36977 
 4:1,?: e1a 36977 
 5:2,3,?: e11 37038 
 6:5,4,?: e11 37038 
 7:?: 
 8:6,7,?: e10 37044 
 9:?: 
 10:8,9,?: e10 37044 
 qed:10: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbc3orgVD 37220 
Virtual deduction proof of sbc3orgOLD 36863. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:: 
 32:3: 
 33:1,32,?: e10 37044 
 4:1,33,?: e11 37038 
 5:2,4,?: e11 37038 
 6:1,?: e1a 36977 
 7:6,?: e1a 36977 
 8:5,7,?: e11 37038 
 9:?: 
 10:8,9,?: e10 37044 
 qed:10: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  19.21a3con13vVD 37221* 
Virtual deduction proof of alrim3con13v 36864. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:2,?: e2 36981 
 4:2,?: e2 36981 
 5:2,?: e2 36981 
 6:1,4,?: e12 37084 
 7:3,?: e2 36981 
 8:5,?: e2 36981 
 9:7,6,8,?: e222 36986 
 10:9,?: e2 36981 
 11:10:in2 
 qed:11:in1 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  exbirVD 37222 
Virtual deduction proof of exbir 36803. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:: 
 5:1,2,?: e12 37084 
 6:3,5,?: e32 37118 
 7:6: 
 8:7: 
 9:8,?: e1a 36977 
 qed:9: 

(Contributed by Alan Sare, 13Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  exbiriVD 37223 
Virtual deduction proof of exbiri 626. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
h1:: 
 2:: 
 3:: 
 4:: 
 5:2,1,?: e10 37044 
 6:3,5,?: e21 37090 
 7:4,6,?: e32 37118 
 8:7: 
 9:8: 
 qed:9: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  rspsbc2VD 37224* 
Virtual deduction proof of rspsbc2 36865. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:: 
 4:1,3,?: e13 37108 
 5:1,4,?: e13 37108 
 6:2,5,?: e23 37115 
 7:6: 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpVD 37225 
Virtual deduction proof of 3impexp 1227. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e10 37044 
 4:3,?: e1a 36977 
 5:4,?: e1a 36977 
 6:5: 
 7:: 
 8:7,?: e1a 36977 
 9:8,?: e1a 36977 
 10:2,9,?: e01 37041 
 11:10: 
 qed:6,11,?: e00 37128 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpbicomVD 37226 
Virtual deduction proof of 3impexpbicom 36804. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e10 37044 
 4:3,?: e1a 36977 
 5:4: 
 6:: 
 7:6,?: e1a 36977 
 8:7,2,?: e10 37044 
 9:8: 
 qed:5,9,?: e00 37128 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3impexpbicomiVD 37227 
Virtual deduction proof of 3impexpbicomi 36805. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcel1gvOLD 37228* 
Class substitution into a membership relation. (Contributed by NM,
17Nov2006.) (Proof shortened by Andrew Salmon, 29Jun2011.)
Obsolete as of 17Aug2018. Use sbcel1v 3358 instead.
(New usage is discouraged.) (Proof modification is discouraged.)



Theorem  sbcoreleleqVD 37229* 
Virtual deduction proof of sbcoreleleq 36866. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:1,?: e1a 36977 
 4:1,?: e1a 36977 
 5:2,3,4,?: e111 37024 
 6:1,?: e1a 36977 
 7:5,6: e11 37038 
 qed:7: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  hbra2VD 37230* 
Virtual deduction proof of nfra2 2809. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
1:: 
 2:: 
 3:1,2,?: e00 37128 
 4:2: 
 5:4,?: e0a 37132 
 qed:3,5,?: e00 37128 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  tratrbVD 37231* 
Virtual deduction proof of tratrb 36867. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:1,?: e1a 36977 
 4:1,?: e1a 36977 
 5:: 
 6:5,?: e2 36981 
 7:5,?: e2 36981 
 8:2,7,4,?: e121 37006 
 9:2,6,8,?: e122 37003 
 10:: 
 11:6,7,10,?: e223 36985 
 12:11: 
 13:: 
 14:12,13,?: e20 37087 
 15:: 
 16:7,15,?: e23 37115 
 17:6,16,?: e23 37115 
 18:17: 
 19:: 
 20:18,19,?: e20 37087 
 21:3,?: e1a 36977 
 22:21,9,4,?: e121 37006 
 23:22,?: e2 36981 
 24:4,23,?: e12 37084 
 25:14,20,24,?: e222 36986 
 26:25: 
 27:: 
 28:27,?: e0a 37132 
 29:28,26: 
 30:: 
 31:30,?: e0a 37132 
 32:31,29: 
 33:32,?: e1a 36977 
 qed:33: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  al2imVD 37232 
Virtual deduction proof of al2im 1680. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:: 
 4:2,3,?: e10 37044 
 qed:4: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  syl5impVD 37233 
Virtual deduction proof of syl5imp 36839. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: 
 2:1,?: e1a 36977 
 3:: 
 4:3,2,?: e21 37090 
 5:4,?: e2 36981 
 6:5: 
 qed:6: 

(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  idiVD 37234 
Virtual deduction proof of idiALT 36802. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ancomstVD 37235 
Closed form of ancoms 454. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
The proof of ancomst 453 is derived automatically from it.
(Contributed by Alan Sare, 25Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ssralv2VD 37236* 
Quantification restricted to a subclass for two quantifiers. ssralv 3525
for two quantifiers. 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. ssralv2 36858 is ssralv2VD 37236 without
virtual deductions and was automatically derived from ssralv2VD 37236.
1:: 
 2:: 
 3:1: 
 4:3,2: 
 5:4: 
 6:5: 
 7:: 
 8:7,6: 
 9:1: 
 10:9,8: 
 11:10: 
 12:: 
 13:: 
 14:12,13,11: 
 15:14: 
 16:15: 
 qed:16: 

(Contributed by Alan Sare, 10Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ordelordALTVD 37237 
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 5464 using
the Axiom of Regularity indirectly through dford2 8134. dford2 is a
weaker definition of ordinal number. Given the Axiom of Regularity, it
need not be assumed that because this is inferred by the
Axiom of Regularity. 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. ordelordALT 36868 is ordelordALTVD 37237
without virtual deductions and was automatically derived from
ordelordALTVD 37237 using the tools program
translate..without..overwriting.cmd and Metamath's minimize command.
1:: 
 2:1: 
 3:1: 
 4:2: 
 5:2: 
 6:4,3: 
 7:6,6,5: 
 8:: 
 9:8: 
 10:9: 
 11:10: 
 12:11: 
 13:12: 
 14:13: 
 15:14,5: 
 16:4,15,3: 
 17:16,7: 
 qed:17: 

(Contributed by Alan Sare, 12Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  equncomVD 37238 
If a class equals the union of two other classes, then it equals the
union of those two classes commuted. 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. equncom 3611 is equncomVD 37238 without
virtual deductions and was automatically derived from equncomVD 37238.
1:: 
 2:: 
 3:1,2: 
 4:3: 
 5:: 
 6:5,2: 
 7:6: 
 8:4,7: 

(Contributed by Alan Sare, 17Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  equncomiVD 37239 
Inference form of equncom 3611. 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. equncomi 3612 is equncomiVD 37239 without
virtual deductions and was automatically derived from equncomiVD 37239.
(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sucidALTVD 37240 
A set belongs to its successor. Alternate proof of sucid 5521.
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. sucidALT 37241 is sucidALTVD 37240
without virtual deductions and was automatically derived from
sucidALTVD 37240. This proof illustrates that
completeusersproof.cmd will generate a Metamath proof from any
User's Proof which is "conventional" in the sense that no step
is a virtual deduction, provided that all necessary unification
theorems and transformation deductions are in set.mm.
completeusersproof.cmd automatically converts such a
conventional proof into a Virtual Deduction proof for which each
step happens to be a 0virtual hypothesis virtual deduction.
The user does not need to search for reference theorem labels or
deduction labels nor does he(she) need to use theorems and
deductions which unify with reference theorems and deductions in
set.mm. All that is necessary is that each theorem or deduction
of the User's Proof unifies with some reference theorem or
deduction in set.mm or is a semantic variation of some theorem
or deduction which unifies with some reference theorem or
deduction in set.mm. The definition of "semantic variation" has
not been precisely defined. If it is obvious that a theorem or
deduction has the same meaning as another theorem or deduction,
then it is a semantic variation of the latter theorem or
deduction. For example, step 4 of the User's Proof is a
semantic variation of the definition (axiom)
, which unifies with dfsuc 5448, a
reference definition (axiom) in set.mm. Also, a theorem or
deduction is said to be a semantic variation of another
theorem or deduction if it is obvious upon cursory inspection
that it has the same meaning as a weaker form of the latter
theorem or deduction. For example, the deduction
infers is a
semantic variation of the theorem
, which unifies with
the set.mm reference definition (axiom) dford2 8134.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sucidALT 37241 
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 37240 using translate_{without}_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18Feb2012.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  sucidVD 37242 
A set belongs to its successor. 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.
sucid 5521 is sucidVD 37242 without virtual deductions and was automatically
derived from sucidVD 37242.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  imbi12VD 37243 
Implication form of imbi12i 327.
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.
imbi12 323 is imbi12VD 37243 without virtual deductions and was automatically
derived from imbi12VD 37243.
1:: 
 2:: 
 3:: 
 4:1,3: 
 5:2,4: 
 6:5: 
 7:: 
 8:1,7: 
 9:2,8: 
 10:9: 
 11:6,10: 
 12:11: 
 qed:12: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  imbi13VD 37244 
Join three logical equivalences to form equivalence of implications.
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.
imbi13 36847 is imbi13VD 37244 without virtual deductions and was automatically
derived from imbi13VD 37244.
1:: 
 2:: 
 3:: 
 4:2,3: 
 5:1,4: 
 6:5: 
 7:6: 
 qed:7: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcim2gVD 37245 
Distribution of class substitution over a leftnested implication.
Similar to sbcimg 3341.
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.
sbcim2g 36869 is sbcim2gVD 37245 without virtual deductions and was automatically
derived from sbcim2gVD 37245.
1:: 
 2:: 
 3:1,2: 
 4:1: 
 5:3,4: 
 6:5: 
 7:: 
 8:4,7: 
 9:1: 
 10:8,9: 
 11:10: 
 12:6,11: 
 qed:12: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcbiVD 37246 
Implication form of sbcbiiOLD 36862.
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.
sbcbi 36870 is sbcbiVD 37246 without virtual deductions and was automatically
derived from sbcbiVD 37246.
1:: 
 2:: 
 3:1,2: 
 4:1,3: 
 5:4: 
 qed:5: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trsbcVD 37247* 
Formulabuilding inference rule for class substitution, substituting a
class variable for the setvar variable of the transitivity predicate.
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.
trsbc 36871 is trsbcVD 37247 without virtual deductions and was automatically
derived from trsbcVD 37247.
1:: 
 2:1: 
 3:1: 
 4:1: 
 5:1,2,3,4: 
 6:1: 
 7:5,6: 
 8:: 
 9:7,8: 
 10:: 
 11:10: 
 12:1,11: 
 13:9,12: 
 14:13: 
 15:14: 
 16:1: 
 17:15,16: 
 18:17: 
 19:18: 
 20:1: 
 21:19,20: 
 22:: 
 23:21,22: 
 24:: 
 25:24: 
 26:1,25: 
 27:23,26: 
 qed:27: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  truniALTVD 37248* 
The union of a class of transitive sets is transitive.
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.
truniALT 36872 is truniALTVD 37248 without virtual deductions and was
automatically derived from truniALTVD 37248.
1:: 
 2:: 
 3:2: 
 4:2: 
 5:4: 
 6:: 
 7:6: 
 8:6: 
 9:1,8: 
 10:8,9: 
 11:3,7,10: 
 12:11,8: 
 13:12: 
 14:13: 
 15:14: 
 16:5,15: 
 17:16: 
 18:17: 
 19:18: 
 qed:19: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ee33VD 37249 
Nonvirtual deduction form of e33 37094.
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.
ee33 36848 is ee33VD 37249 without virtual deductions and was automatically
derived from ee33VD 37249.
h1:: 
 h2:: 
 h3:: 
 4:1,3: 
 5:4: 
 6:2,5: 
 7:6: 
 8:7: 
 qed:8: 

(Contributed by Alan Sare, 18Mar2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trintALTVD 37250* 
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 37251.
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 37251 is trintALTVD 37250 without virtual deductions and was
automatically derived from trintALTVD 37250.
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, 17Apr2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  trintALT 37251* 
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 37251 is an alternate proof of trint 4533.
trintALT 37251 is trintALTVD 37250 without virtual deductions and was
automatically derived from trintALTVD 37250 using the tools program
translate..without..overwriting.cmd and Metamath's minimize command.
(Contributed by Alan Sare, 17Apr2012.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  undif3VD 37252 
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 3734.
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.
undif3 3734 is undif3VD 37252 without virtual deductions and was automatically
derived from undif3VD 37252.
1:: 
 2:: 
 3:2: 
 4:1,3: 
 5:: 
 6:5: 
 7:5: 
 8:6,7: 
 9:8: 
 10:: 
 11:10: 
 12:10: 
 13:11: 
 14:12: 
 15:13,14: 
 16:15: 
 17:9,16: 
 18:: 
 19:18: 
 20:18: 
 21:18: 
 22:21: 
 23:: 
 24:23: 
 25:24: 
 26:25: 
 27:10: 
 28:27: 
 29:: 
 30:29: 
 31:30: 
 32:31: 
 33:22,26: 
 34:28,32: 
 35:33,34: 
 36:: 
 37:36,35: 
 38:17,37: 
 39:: 
 40:39: 
 41:: 
 42:40,41: 
 43:: 
 44:43,42: 
 45:: 
 46:45,44: 
 47:4,38: 
 48:46,47: 
 49:48: 
 qed:49: 

(Contributed by Alan Sare, 17Apr2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sbcssgVD 37253 
Virtual deduction proof of sbcssg 3910.
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.
sbcssg 3910 is sbcssgVD 37253 without virtual deductions and was automatically
derived from sbcssgVD 37253.
1:: 
 2:1: 
 3:1: 
 4:2,3: 
 5:1: 
 6:4,5: 
 7:6: 
 8:7: 
 9:1: 
 10:8,9: 
 11:: 
 110:11: 
 12:1,110: 
 13:10,12: 
 14:: 
 15:13,14: 
 qed:15: 

(Contributed by Alan Sare, 22Jul2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  csbingVD 37254 
Virtual deduction proof of csbingOLD 37188.
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.
csbingOLD 37188 is csbingVD 37254 without virtual deductions and was
automatically derived from csbingVD 37254.
1:: 
 2:: 
 20:2: 
 30:1,20: 
 3:1,30: 
 4:1: 
 5:3,4: 
 6:1: 
 7:1: 
 8:6,7: 
 9:1: 
 10:9,8: 
 11:10: 
 12:11: 
 13:5,12: 
 14:: 
 15:13,14: 
 qed:15: 

(Contributed by Alan Sare, 22Jul2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  onfrALTlem5VD 37255* 
Virtual deduction proof of onfrALTlem5 36878.
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.
onfrALTlem5 36878 is onfrALTlem5VD 37255 without virtual deductions and was
automatically derived from onfrALTlem5VD 37255.
1:: 
 2:1: 
 3:2: 
 4:3: 
 5:: 
 6:4,5: 
 7:2: 
 8:: 
 9:8: 
 10:2,9: 
 11:7,10: 
 12:6,11: 
 13:2: 
 14:12,13: 
 15:2: 
 16:15,14: 
 17:2: 
 18:2: 
 19:2: 
 20:18,19: 
 21:17,20: 
 22:2: 
 23:2: 
 24:21,23: 
 25:22,24: 
 26:2: 
 27:25,26: 
 28:2: 
 29:27,28: 
 30:29: 
 31:30: 
 32:: 
 33:31,32: 
 34:2: 
 35:33,34: 
 36:: 
 37:36: 
 38:2,37: 
 39:35,38: 
 40:16,39: 
 41:2: 
 qed:40,41: 

(Contributed by Alan Sare, 22Jul2012.) (Proof modification is
discouraged.) (New usage is discouraged.)

