Type  Label  Description 
Statement 

Theorem  unipwr 33501 
A class is a subclass of the union of its power class. This theorem is
the righttoleft subclass lemma of unipw 4687. The proof of this theorem
was automatically generated from unipwrVD 33500 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 33502 
Virtual deduction proof of sstrALT2 33503. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  sstrALT2 33503 
Virtual deduction proof of sstr 3497, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 33502 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 33504 
Virtual deduction proof of suctrALT2 33505. (Contributed by Alan Sare,
11Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  suctrALT2 33505 
Virtual deduction proof of suctr 4951. The sucessor of a transitive class
is transitive. This proof was generated automatically from the virtual
deduction proof suctrALT2VD 33504 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 33506* 
Virtual deduction proof of elex2 3107. (Contributed by Alan Sare,
25Sep2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  elex22VD 33507* 
Virtual deduction proof of elex22 3108. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  eqsbc3rVD 33508* 
Virtual deduction proof of eqsbc3r 3375. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  zfregs2VD 33509* 
Virtual deduction proof of zfregs2 8167. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  tpid3gVD 33510 
Virtual deduction proof of tpid3g 4130. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem1VD 33511* 
Virtual deduction proof of en3lplem1 8034. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lplem2VD 33512* 
Virtual deduction proof of en3lplem2 8035. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  en3lpVD 33513 
Virtual deduction proof of en3lp 8036. (Contributed by Alan Sare,
24Oct2011.) (Proof modification is discouraged.)
(New usage is discouraged.)



21.27.8 Theorems proved using Virtual Deduction
with mmj2 assistance


Theorem  simplbi2VD 33514 
Virtual deduction proof of simplbi2 625. 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 33437 
 qed:3,?: e0a 33437 

The proof of simplbi2 625 was automatically derived from it.
(Contributed by Alan Sare, 31Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  3ornot23VD 33515 
Virtual deduction proof of 3ornot23 33146. 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 33281 
 4:1,?: e1a 33281 
 5:3,4,?: e11 33342 
 6:2,?: e2 33285 
 7:5,6,?: e12 33389 
 8:7: 
 qed:8: 

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



Theorem  orbi1rVD 33516 
Virtual deduction proof of orbi1r 33147. 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 33285 
 4:1,3,?: e12 33389 
 5:4,?: e2 33285 
 6:5: 
 7:: 
 8:7,?: e2 33285 
 9:1,8,?: e12 33389 
 10:9,?: e2 33285 
 11:10: 
 12:6,11,?: e11 33342 
 qed:12: 

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



Theorem  bitr3VD 33517 
Virtual deduction proof of bitr3 33148. 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 33281 
 3:: 
 4:3,?: e2 33285 
 5:2,4,?: e12 33389 
 6:5: 
 qed:6: 

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



Theorem  3orbi123VD 33518 
Virtual deduction proof of 3orbi123 33149. 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 33281 
 3:1,?: e1a 33281 
 4:1,?: e1a 33281 
 5:2,3,?: e11 33342 
 6:5,4,?: e11 33342 
 7:?: 
 8:6,7,?: e10 33348 
 9:?: 
 10:8,9,?: e10 33348 
 qed:10: 

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



Theorem  sbc3orgVD 33519 
Virtual deduction proof of sbc3orgOLD 33171. 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 33281 
 3:: 
 32:3: 
 33:1,32,?: e10 33348 
 4:1,33,?: e11 33342 
 5:2,4,?: e11 33342 
 6:1,?: e1a 33281 
 7:6,?: e1a 33281 
 8:5,7,?: e11 33342 
 9:?: 
 10:8,9,?: e10 33348 
 qed:10: 

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



Theorem  19.21a3con13vVD 33520* 
Virtual deduction proof of alrim3con13v 33172. 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 33285 
 4:2,?: e2 33285 
 5:2,?: e2 33285 
 6:1,4,?: e12 33389 
 7:3,?: e2 33285 
 8:5,?: e2 33285 
 9:7,6,8,?: e222 33290 
 10:9,?: e2 33285 
 11:10:in2 
 qed:11:in1 

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



Theorem  exbirVD 33521 
Virtual deduction proof of exbir 33087. 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 33389 
 6:3,5,?: e32 33423 
 7:6: 
 8:7: 
 9:8,?: e1a 33281 
 qed:9: 

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



Theorem  exbiriVD 33522 
Virtual deduction proof of exbiri 622. 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 33348 
 6:3,5,?: e21 33395 
 7:4,6,?: e32 33423 
 8:7: 
 9:8: 
 qed:9: 

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



Theorem  rspsbc2VD 33523* 
Virtual deduction proof of rspsbc2 33173. 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 33413 
 5:1,4,?: e13 33413 
 6:2,5,?: e23 33420 
 7:6: 
 8:7: 
 qed:8: 

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



Theorem  3impexpVD 33524 
Virtual deduction proof of 3impexp 33088. 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 33348 
 4:3,?: e1a 33281 
 5:4,?: e1a 33281 
 6:5: 
 7:: 
 8:7,?: e1a 33281 
 9:8,?: e1a 33281 
 10:2,9,?: e01 33345 
 11:10: 
 qed:6,11,?: e00 33433 

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



Theorem  3impexpbicomVD 33525 
Virtual deduction proof of 3impexpbicom 33089. 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 33348 
 4:3,?: e1a 33281 
 5:4: 
 6:: 
 7:6,?: e1a 33281 
 8:7,2,?: e10 33348 
 9:8: 
 qed:5,9,?: e00 33433 

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



Theorem  3impexpbicomiVD 33526 
Virtual deduction proof of 3impexpbicomi 33090. 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  sbcoreleleqVD 33527* 
Virtual deduction proof of sbcoreleleq 33174. 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 33281 
 3:1,?: e1a 33281 
 4:1,?: e1a 33281 
 5:2,3,4,?: e111 33328 
 6:1,?: e1a 33281 
 7:5,6: e11 33342 
 qed:7: 

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



Theorem  hbra2VD 33528* 
Virtual deduction proof of nfra2 2830. 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 33433 
 4:2: 
 5:4,?: e0a 33437 
 qed:3,5,?: e00 33433 

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



Theorem  tratrbVD 33529* 
Virtual deduction proof of tratrb 33175. 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 33281 
 3:1,?: e1a 33281 
 4:1,?: e1a 33281 
 5:: 
 6:5,?: e2 33285 
 7:5,?: e2 33285 
 8:2,7,4,?: e121 33310 
 9:2,6,8,?: e122 33307 
 10:: 
 11:6,7,10,?: e223 33289 
 12:11: 
 13:: 
 14:12,13,?: e20 33392 
 15:: 
 16:7,15,?: e23 33420 
 17:6,16,?: e23 33420 
 18:17: 
 19:: 
 20:18,19,?: e20 33392 
 21:3,?: e1a 33281 
 22:21,9,4,?: e121 33310 
 23:22,?: e2 33285 
 24:4,23,?: e12 33389 
 25:14,20,24,?: e222 33290 
 26:25: 
 27:: 
 28:27,?: e0a 33437 
 29:28,26: 
 30:: 
 31:30,?: e0a 33437 
 32:31,29: 
 33:32,?: e1a 33281 
 qed:33: 

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



Theorem  al2imVD 33530 
Virtual deduction proof of al2im 1622. 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 33281 
 3:: 
 4:2,3,?: e10 33348 
 qed:4: 

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



Theorem  syl5impVD 33531 
Virtual deduction proof of syl5imp 33150. 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 33281 
 3:: 
 4:3,2,?: e21 33395 
 5:4,?: e2 33285 
 6:5: 
 qed:6: 

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



Theorem  idiVD 33532 
Virtual deduction proof of idiALT 33086. 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 33533 
Closed form of ancoms 453. 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 452 is derived automatically from it.
(Contributed by Alan Sare, 25Dec2011.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  ssralv2VD 33534* 
Quantification restricted to a subclass for two quantifiers. ssralv 3549
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 33169 is ssralv2VD 33534 without
virtual deductions and was automatically derived from ssralv2VD 33534.
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 33535 
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 4890 using
the Axiom of Regularity indirectly through dford2 8040. 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 33176 is ordelordALTVD 33535
without virtual deductions and was automatically derived from
ordelordALTVD 33535 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 33536 
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 3634 is equncomVD 33536 without
virtual deductions and was automatically derived from equncomVD 33536.
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 33537 
Inference form of equncom 3634. 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 3635 is equncomiVD 33537 without
virtual deductions and was automatically derived from equncomiVD 33537.
(Contributed by Alan Sare, 18Feb2012.) (Proof modification is
discouraged.) (New usage is discouraged.)



Theorem  sucidALTVD 33538 
A set belongs to its successor. Alternate proof of sucid 4947.
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 33539 is sucidALTVD 33538
without virtual deductions and was automatically derived from
sucidALTVD 33538. 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 4874, 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 8040.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

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



Theorem  sucidALT 33539 
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 33538 using translate_{without}_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18Feb2012.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  sucidVD 33540 
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 4947 is sucidVD 33540 without virtual deductions and was automatically
derived from sucidVD 33540.
h1:: 
 2:1: 
 3:2: 
 4:: 
 qed:3,4: 

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



Theorem  imbi12VD 33541 
Implication form of imbi12i 326.
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 322 is imbi12VD 33541 without virtual deductions and was automatically
derived from imbi12VD 33541.
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 33542 
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 33158 is imbi13VD 33542 without virtual deductions and was automatically
derived from imbi13VD 33542.
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 33543 
Distribution of class substitution over a leftnested implication.
Similar to sbcimg 3355.
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 33177 is sbcim2gVD 33543 without virtual deductions and was automatically
derived from sbcim2gVD 33543.
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 33544 
Implication form of sbcbiiOLD 3374.
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 33178 is sbcbiVD 33544 without virtual deductions and was automatically
derived from sbcbiVD 33544.
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 33545* 
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 33179 is trsbcVD 33545 without virtual deductions and was automatically
derived from trsbcVD 33545.
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 33546* 
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 33180 is truniALTVD 33546 without virtual deductions and was
automatically derived from truniALTVD 33546.
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 33547 
Nonvirtual deduction form of e33 33399.
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 33159 is ee33VD 33547 without virtual deductions and was automatically
derived from ee33VD 33547.
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 33548* 
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 33549.
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 33549 is trintALTVD 33548 without virtual deductions and was
automatically derived from trintALTVD 33548.
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 33549* 
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 33549 is an alternative proof of
trint 4545. trintALT 33549 is trintALTVD 33548 without virtual deductions and was
automatically derived from trintALTVD 33548 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 33550 
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 3744.
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 3744 is undif3VD 33550 without virtual deductions and was automatically
derived from undif3VD 33550.
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 33551 
Virtual deduction proof of sbcssg 3925.
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 3925 is sbcssgVD 33551 without virtual deductions and was automatically
derived from sbcssgVD 33551.
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 33552 
Virtual deduction proof of csbingOLD 3847.
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 3847 is csbingVD 33552 without virtual deductions and was
automatically derived from csbingVD 33552.
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 33553* 
Virtual deduction proof of onfrALTlem5 33182.
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 33182 is onfrALTlem5VD 33553 without virtual deductions and was
automatically derived from onfrALTlem5VD 33553.
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.)



Theorem  onfrALTlem4VD 33554* 
Virtual deduction proof of onfrALTlem4 33183.
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.
onfrALTlem4 33183 is onfrALTlem4VD 33554 without virtual deductions and was
automatically derived from onfrALTlem4VD 33554.
